1
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Fan H, Wang Y, Du Y, Qiu H, Wang X. Scalable synchronization cluster in networked chaotic oscillators. CHAOS (WOODBURY, N.Y.) 2024; 34:071102. [PMID: 38953751 DOI: 10.1063/5.0218294] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2024] [Accepted: 06/13/2024] [Indexed: 07/04/2024]
Abstract
Cluster synchronization in synthetic networks of coupled chaotic oscillators is investigated. It is found that despite the asymmetric nature of the network structure, a subset of the oscillators can be synchronized as a cluster while the other oscillators remain desynchronized. Interestingly, with the increase in the coupling strength, the cluster is expanding gradually by recruiting the desynchronized oscillators one by one. This new synchronization phenomenon, which is named "scalable synchronization cluster," is explored theoretically by the method of eigenvector-based analysis, and it is revealed that the scalability of the cluster is attributed to the unique feature of the eigenvectors of the network coupling matrix. The transient dynamics of the cluster in response to random perturbations are also studied, and it is shown that in restoring to the synchronization state, oscillators inside the cluster are stabilized in sequence, illustrating again the hierarchy of the oscillators. The findings shed new light on the collective behaviors of networked chaotic oscillators and are helpful for the design of real-world networks where scalable synchronization clusters are concerned.
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Affiliation(s)
- Huawei Fan
- School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
| | - Yafeng Wang
- Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721016, China
| | - Yao Du
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Haibo Qiu
- School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, China
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
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2
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Muthanna YA, Jafri HH. Explosive transitions in coupled Lorenz oscillators. Phys Rev E 2024; 109:054206. [PMID: 38907430 DOI: 10.1103/physreve.109.054206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2023] [Accepted: 04/16/2024] [Indexed: 06/24/2024]
Abstract
We study the transition to synchronization in an ensemble of chaotic oscillators that are interacting on a star network. These oscillators possess an invariant symmetry and we study emergent behavior by introducing the timescale variations in the dynamics of the nodes and the hub. If the coupling preserves the symmetry, the ensemble exhibits consecutive explosive transitions, each one associated with a hysteresis. The first transition is the explosive synchronization from a desynchronized state to a synchronized state which occurs discontinuously with the formation of intermediate clusters. These clusters appear because of the driving-induced multistability and the resulting attractors exhibit intermittent synchrony (antisynchrony). The second transition is the explosive death that occurs as a result of stabilization of the stable fixed points. However, if the symmetry is not preserved, the system again makes a first-order transition from an oscillatory state to death, namely, an explosive death. These transitions are studied with the help of the master stability functions, Lyapunov exponents, and the stability analysis.
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Affiliation(s)
- Yusra Ahmed Muthanna
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
- Physics Department, Taiz University, Taiz 6803, Yemen
| | - Haider Hasan Jafri
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
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3
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Ghosh S, Pal S, Sar GK, Ghosh D. Amplitude responses of swarmalators. Phys Rev E 2024; 109:054205. [PMID: 38907391 DOI: 10.1103/physreve.109.054205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2024] [Accepted: 04/17/2024] [Indexed: 06/24/2024]
Abstract
Swarmalators are entities that swarm through space and sync in time and are potentially considered to replicate the complex dynamics of many real-world systems. So far, the internal dynamics of swarmalators have been taken as a phase oscillator inspired by the Kuramoto model. Here we examine the internal dynamics utilizing an amplitude oscillator capable of exhibiting periodic and chaotic behaviors. To incorporate the dual interplay between spatial and internal dynamics, we propose a general model that keeps the properties of swarmalators intact. This adaptation calls for a detailed study, which we present in this paper. We establish our study with the Rössler oscillator by taking parameters from both chaotic and periodic regions. While the periodic oscillator mimics most of the patterns in the previous phase oscillator model, the chaotic oscillator brings some fascinating states.
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Affiliation(s)
- Samali Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Suvam Pal
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Gourab Kumar Sar
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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4
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Aristides RP, Cerdeira HA. Master stability functions of networks of Izhikevich neurons. Phys Rev E 2024; 109:044213. [PMID: 38755844 DOI: 10.1103/physreve.109.044213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2022] [Accepted: 04/02/2024] [Indexed: 05/18/2024]
Abstract
Synchronization has attracted interest in many areas where the systems under study can be described by complex networks. Among such areas is neuroscience, where it is hypothesized that synchronization plays a role in many functions and dysfunctions of the brain. We study the linear stability of synchronized states in networks of Izhikevich neurons using master stability functions (MSFs), and to accomplish that, we exploit the formalism of saltation matrices. Such a tool allows us to calculate the Lyapunov exponents of the MSF properly since the Izhikevich model displays a discontinuity within its spikes. We consider both electrical and chemical couplings as well as global and cluster synchronized states. The MSF calculations are compared with a measure of the synchronization error for simulated networks. We give special attention to the case of electric and chemical coupling, where a riddled basin of attraction makes the synchronized solution more sensitive to perturbations.
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Affiliation(s)
- Raul P Aristides
- São Paulo State University (UNESP), Instituto de Física Teórica, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil
| | - Hilda A Cerdeira
- São Paulo State University (UNESP), Instituto de Física Teórica, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil. and Epistemic, Gómez & Gómez Ltda. ME, Rua Paulo Franco 520, Vila Leopoldina, 05305-031 São Paulo, Brazil
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5
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Papo D, Buldú JM. Does the brain behave like a (complex) network? I. Dynamics. Phys Life Rev 2024; 48:47-98. [PMID: 38145591 DOI: 10.1016/j.plrev.2023.12.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2023] [Accepted: 12/10/2023] [Indexed: 12/27/2023]
Abstract
Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.
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Affiliation(s)
- D Papo
- Department of Neuroscience and Rehabilitation, Section of Physiology, University of Ferrara, Ferrara, Italy; Center for Translational Neurophysiology, Fondazione Istituto Italiano di Tecnologia, Ferrara, Italy.
| | - J M Buldú
- Complex Systems Group & G.I.S.C., Universidad Rey Juan Carlos, Madrid, Spain
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6
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Irankhah R, Mehrabbeik M, Parastesh F, Rajagopal K, Jafari S, Kurths J. Synchronization enhancement subjected to adaptive blinking coupling. CHAOS (WOODBURY, N.Y.) 2024; 34:023120. [PMID: 38377293 DOI: 10.1063/5.0188366] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2023] [Accepted: 01/25/2024] [Indexed: 02/22/2024]
Abstract
Synchronization holds a significant role, notably within chaotic systems, in various contexts where the coordinated behavior of systems plays a pivotal and indispensable role. Hence, many studies have been dedicated to investigating the underlying mechanism of synchronization of chaotic systems. Networks with time-varying coupling, particularly those with blinking coupling, have been proven essential. The reason is that such coupling schemes introduce dynamic variations that enhance adaptability and robustness, making them applicable in various real-world scenarios. This paper introduces a novel adaptive blinking coupling, wherein the coupling adapts dynamically based on the most influential variable exhibiting the most significant average disparity. To ensure an equitable selection of the most effective coupling at each time instance, the average difference of each variable is normalized to the synchronous solution's range. Due to this adaptive coupling selection, synchronization enhancement is expected to be observed. This hypothesis is assessed within networks of identical systems, encompassing Lorenz, Rössler, Chen, Hindmarsh-Rose, forced Duffing, and forced van der Pol systems. The results demonstrated a substantial improvement in synchronization when employing adaptive blinking coupling, particularly when applying the normalization process.
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Affiliation(s)
- Reza Irankhah
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 159163-4311, Iran
| | - Mahtab Mehrabbeik
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 159163-4311, Iran
| | - Fatemeh Parastesh
- Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, India
| | - Karthikeyan Rajagopal
- Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, India
| | - Sajad Jafari
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 159163-4311, Iran
- Health Technology Research Institute, Amirkabir University of Technology (Tehran Polytechnic), Tehran 159163-4311, Iran
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany
- Institute of Physics, Humboldt University of Berlin, Berlin 12489, Germany
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7
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Nicola W. Rapid changes in synchronizability in conductance-based neuronal networks with conductance-based coupling. CHAOS (WOODBURY, N.Y.) 2024; 34:023123. [PMID: 38377288 DOI: 10.1063/5.0176956] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/18/2023] [Accepted: 01/14/2024] [Indexed: 02/22/2024]
Abstract
Real neurons connect to each other non-randomly. These connectivity graphs can potentially impact the ability of networks to synchronize, along with the dynamics of neurons and the dynamics of their connections. How the connectivity of networks of conductance-based neuron models like the classical Hodgkin-Huxley model or the Morris-Lecar model impacts synchronizability remains unknown. One powerful tool to resolve the synchronizability of these networks is the master stability function (MSF). Here, we apply and extend the MSF approach to networks of Morris-Lecar neurons with conductance-based coupling to determine under which parameters and for which graphs the synchronous solutions are stable. We consider connectivity graphs with a constant non-zero row sum, where the MSF approach can be readily extended to conductance-based synapses rather than the more well-studied diffusive connectivity case, which primarily applies to gap junction connectivity. In this formulation, the synchronous solution is a single, self-coupled, or "autaptic" neuron. We find that the primary determining parameter for the stability of the synchronous solution is, unsurprisingly, the reversal potential, as it largely dictates the excitatory/inhibitory potential of a synaptic connection. However, the change between "excitatory" and "inhibitory" synapses is rapid, with only a few millivolts separating stability and instability of the synchronous state for most graphs. We also find that for specific coupling strengths (as measured by the global synaptic conductance), islands of synchronizability in the MSF can emerge for inhibitory connectivity. We verified the stability of these islands by direct simulation of pairs of neurons coupled with eigenvalues in the matching spectrum.
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8
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Joseph D, Kumar R, Karthikeyan A, Rajagopal K. Dynamics, synchronization and traveling wave patterns of flux coupled network of Chay neurons. Biosystems 2024; 235:105113. [PMID: 38159671 DOI: 10.1016/j.biosystems.2023.105113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2023] [Revised: 12/22/2023] [Accepted: 12/24/2023] [Indexed: 01/03/2024]
Abstract
Studies in the literature have demonstrated the significance of the synchronization of neuronal electrical activity for signal transmission and information encoding. In light of this importance, we investigate the synchronization of the Chay neuron model using both theoretical analysis and numerical simulations. The Chay model is chosen for its comprehensive understanding of neuronal behavior and computational efficiency. Additionally, we explore the impact of electromagnetic induction, leading to the magnetic flux Chay neuron model. The single neuron model exhibits rich and complex dynamics for various parameter choices. We explore the bifurcation structure of the model through bifurcation diagrams and Lyapunov exponents. Subsequently, we extend our study to two coupled magnetic flux Chay neurons, identifying mode locking and structures reminiscent of Arnold's tongue. We evaluate the stability of the synchronized manifold using Lyapunov theory and confirm our findings through simulations. Expanding our study to networks of diffusively coupled flux Chay neurons, we observe coherent, incoherent, and imperfect chimera patterns. Our investigation of three network types highlights the impact of network topology on the emergent dynamics of the Chay neuron network. Regular networks exhibit diverse patterns, small-world networks demonstrate a critical transition to coherence, and random networks showcase synchronization at specific coupling strengths. These findings significantly contribute to our understanding of the synchronization patterns exhibited by the magnetic flux Chay neuron. To assess the synchronization stability of the Chay neuron network, we employ master stability function analysis.
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Affiliation(s)
- Dianavinnarasi Joseph
- Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, India.
| | - Rakshanaa Kumar
- Department of Information Technology, Chennai Institute of Technology, Chennai 600069, India.
| | - Anitha Karthikeyan
- Department of Electronics and Communication Engineering, Vemu Institute of Technology, Chitoor, Andhra Pradesh 517112, India; Department of Electronics and Communications Engineering and University Centre for Research & Development, Chandigarh University, Mohali 140413, India.
| | - Karthikeyan Rajagopal
- Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, India.
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9
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Brice Azangue A, Megam Ngouonkadi EB, Kabong Nono M, Fotsin HB, Sone Ekonde M, Yemele D. Stability and synchronization in neural network with delayed synaptic connections. CHAOS (WOODBURY, N.Y.) 2024; 34:013117. [PMID: 38215223 DOI: 10.1063/5.0175408] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2023] [Accepted: 12/04/2023] [Indexed: 01/14/2024]
Abstract
In this paper, we investigate the stability of the synchronous state in a complex network using the master stability function technique. We use the extended Hindmarsh-Rose neuronal model including time delayed electrical, chemical, and hybrid couplings. We find the corresponding master stability equation that describes the whole dynamics for each coupling mode. From the maximum Lyapunov exponent, we deduce the stability state for each coupling mode. We observe that for electrical coupling, there exists a mixing between stable and unstable states. For a good setting of some system parameters, the position and the size of unstable areas can be modified. For chemical coupling, we observe difficulties in having a stable area in the complex plane. For hybrid coupling, we observe a stable behavior in the whole system compared to the case where these couplings are considered separately. The obtained results for each coupling mode help to analyze the stability state of some network topologies by using the corresponding eigenvalues. We observe that using electrical coupling can involve a full or partial stability of the system. In the case of chemical coupling, unstable states are observed whereas in the case of hybrid interactions a full stability of the network is obtained. Temporal analysis of the global synchronization is also done for each coupling mode, and the results show that when the network is stable, the synchronization is globally observed, while in the case when it is unstable, its nodes are not globally synchronized.
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Affiliation(s)
- A Brice Azangue
- Research Unit of Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 067 Dschang, Cameroon
| | - E B Megam Ngouonkadi
- Research Unit of Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 067 Dschang, Cameroon
- Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63 Buea, Cameroon
| | - M Kabong Nono
- Research Unit of Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 067 Dschang, Cameroon
| | - H B Fotsin
- Research Unit of Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 067 Dschang, Cameroon
| | - M Sone Ekonde
- Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63 Buea, Cameroon
| | - D Yemele
- Research Unit of Mechanics and Modeling of Physical Systems, Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 067 Dschang, Cameroon
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10
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Ren X, Lei Y, Grebogi C, Baptista MS. The complementary contribution of each order topology into the synchronization of multi-order networks. CHAOS (WOODBURY, N.Y.) 2023; 33:111101. [PMID: 37909900 DOI: 10.1063/5.0177687] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2023] [Accepted: 10/12/2023] [Indexed: 11/03/2023]
Abstract
Higher-order interactions improve our capability to model real-world complex systems ranging from physics and neuroscience to economics and social sciences. There is great interest nowadays in understanding the contribution of higher-order terms to the collective behavior of the network. In this work, we investigate the stability of complete synchronization of complex networks with higher-order structures. We demonstrate that the synchronization level of a network composed of nodes interacting simultaneously via multiple orders is maintained regardless of the intensity of coupling strength across different orders. We articulate that lower-order and higher-order topologies work together complementarily to provide the optimal stable configuration, challenging previous conclusions that higher-order interactions promote the stability of synchronization. Furthermore, we find that simply adding higher-order interactions based on existing connections, as in simple complexes, does not have a significant impact on synchronization. The universal applicability of our work lies in the comprehensive analysis of different network topologies, including hypergraphs and simplicial complexes, and the utilization of appropriate rescaling to assess the impact of higher-order interactions on synchronization stability.
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Affiliation(s)
- Xiaomin Ren
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Youming Lei
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| | - Murilo S Baptista
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
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11
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Nazerian A, Nathe C, Hart JD, Sorrentino F. Synchronizing chaos using reservoir computing. CHAOS (WOODBURY, N.Y.) 2023; 33:103121. [PMID: 37832520 PMCID: PMC10576628 DOI: 10.1063/5.0161076] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2023] [Accepted: 09/14/2023] [Indexed: 10/15/2023]
Abstract
We attempt to achieve complete synchronization between a drive system unidirectionally coupled with a response system, under the assumption that limited knowledge on the states of the drive is available at the response. Machine-learning techniques have been previously implemented to estimate the states of a dynamical system from limited measurements. We consider situations in which knowledge of the non-measurable states of the drive system is needed in order for the response system to synchronize with the drive. We use a reservoir computer to estimate the non-measurable states of the drive system from its measured states and then employ these measured states to achieve complete synchronization of the response system with the drive.
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Affiliation(s)
- Amirhossein Nazerian
- Mechanical Engineering Department, University of New Mexico, Albuquerque, New Mexico 87131, USA
| | - Chad Nathe
- Mechanical Engineering Department, University of New Mexico, Albuquerque, New Mexico 87131, USA
| | | | - Francesco Sorrentino
- Mechanical Engineering Department, University of New Mexico, Albuquerque, New Mexico 87131, USA
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12
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Hart JD. Estimating the master stability function from the time series of one oscillator via reservoir computing. Phys Rev E 2023; 108:L032201. [PMID: 37849160 DOI: 10.1103/physreve.108.l032201] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2023] [Accepted: 08/10/2023] [Indexed: 10/19/2023]
Abstract
The master stability function (MSF) yields the stability of the globally synchronized state of a network of identical oscillators in terms of the eigenvalues of the adjacency matrix. In order to compute the MSF, one must have an accurate model of an uncoupled oscillator, but often such a model does not exist. We present a reservoir computing technique for estimating the MSF given only the time series of a single, uncoupled oscillator. We demonstrate the generality of our technique by considering a variety of coupling configurations of networks consisting of Lorenz oscillators or Hénon maps.
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Affiliation(s)
- Joseph D Hart
- U.S. Naval Research Laboratory, Code 5675, Washington, DC 20375, USA
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13
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Letellier C, Sendiña-Nadal I, Leyva I, Barbot JP. Generalized synchronization mediated by a flat coupling between structurally nonequivalent chaotic systems. CHAOS (WOODBURY, N.Y.) 2023; 33:093117. [PMID: 37703476 DOI: 10.1063/5.0156025] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 08/14/2023] [Indexed: 09/15/2023]
Abstract
Synchronization of chaotic systems is usually investigated for structurally equivalent systems typically coupled through linear diffusive functions. Here, we focus on a particular type of coupling borrowed from a nonlinear control theory and based on the optimal placement of a sensor-a device measuring the chosen variable-and an actuator-a device applying the actuating (control) signal to a variable's derivative-in the response system, leading to the so-called flat control law. We aim to investigate the dynamics produced by a response system that is flat coupled to a drive system and to determine the degree of generalized synchronization between them using statistical and topological arguments. The general use of a flat control law for getting generalized synchronization is discussed.
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Affiliation(s)
- Christophe Letellier
- Rouen Normandie University-CORIA, Avenue de l'Université, F-76800 Saint-Etienne du Rouvray, France
| | - Irene Sendiña-Nadal
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - I Leyva
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - Jean-Pierre Barbot
- QUARTZ EA7393 Laboratory, ENSEA, 6 Avenue du Ponceau, 95014 Cergy-Pontoise, France
- LS2N, UMR 6004 CNRS, École Centrale de Nantes, 1 rue de la Noë, 44300 Nantes, France
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14
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Zhang M, Yang Y, Yang J. Hierarchy of partially synchronous states in a ring of coupled identical oscillators. Phys Rev E 2023; 108:034202. [PMID: 37849175 DOI: 10.1103/physreve.108.034202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Accepted: 08/25/2023] [Indexed: 10/19/2023]
Abstract
In coupled identical oscillators, complete synchronization has been well formulated; however, partial synchronization still calls for a general theory. In this work, we study the partial synchronization in a ring of N locally coupled identical oscillators. We first establish the correspondence between partially synchronous states and conjugacy classes of subgroups of the dihedral group D_{N}. Then we present a systematic method to identify all partially synchronous dynamics on their synchronous manifolds by reducing a ring of oscillators to short chains with various boundary conditions. We find that partially synchronous states are organized into a hierarchical structure and, along a directed path in the structure, upstream partially synchronous states are less synchronous than downstream ones.
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Affiliation(s)
- Mei Zhang
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
| | - Yuhe Yang
- School of Mathematics, Peking University, Beijing 100871, People's Republic of China
| | - Junzhong Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
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15
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Kong LW, Lai YC. Short-lived chimera states. CHAOS (WOODBURY, N.Y.) 2023; 33:2894496. [PMID: 37276573 DOI: 10.1063/5.0145573] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2023] [Accepted: 04/21/2023] [Indexed: 06/07/2023]
Abstract
In the classic Kuramoto system of coupled two-dimensional rotators, chimera states characterized by the coexistence of synchronous and asynchronous groups of oscillators are long-lived because the average lifetime of these states increases exponentially with the system size. Recently, it was discovered that, when the rotators in the Kuramoto model are three-dimensional, the chimera states become short-lived in the sense that their lifetime scales with only the logarithm of the dimension-augmenting perturbation. We introduce transverse-stability analysis to understand the short-lived chimera states. In particular, on the unit sphere representing three-dimensional (3D) rotations, the long-lived chimera states in the classic Kuramoto system occur on the equator, to which latitudinal perturbations that make the rotations 3D are transverse. We demonstrate that the largest transverse Lyapunov exponent calculated with respect to these long-lived chimera states is typically positive, making them short-lived. The transverse-stability analysis turns the previous numerical scaling law of the transient lifetime into an exact formula: the "free" proportional constant in the original scaling law can now be precisely determined in terms of the largest transverse Lyapunov exponent. Our analysis reinforces the speculation that in physical systems, chimera states can be short-lived as they are vulnerable to any perturbations that have a component transverse to the invariant subspace in which they live.
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Affiliation(s)
- Ling-Wei Kong
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
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16
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Topal I, Eroglu D. Reconstructing Network Dynamics of Coupled Discrete Chaotic Units from Data. PHYSICAL REVIEW LETTERS 2023; 130:117401. [PMID: 37001085 DOI: 10.1103/physrevlett.130.117401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2022] [Revised: 02/17/2023] [Accepted: 02/21/2023] [Indexed: 06/19/2023]
Abstract
Reconstructing network dynamics from data is crucial for predicting the changes in the dynamics of complex systems such as neuron networks; however, previous research has shown that the reconstruction is possible under strong constraints such as the need for lengthy data or small system size. Here, we present a recovery scheme blending theoretical model reduction and sparse recovery to identify the governing equations and the interactions of weakly coupled chaotic maps on complex networks, easing unrealistic constraints for real-world applications. Learning dynamics and connectivity lead to detecting critical transitions for parameter changes. We apply our technique to realistic neuronal systems with and without noise on a real mouse neocortex and artificial networks.
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Affiliation(s)
- Irem Topal
- Faculty of Engineering and Natural Sciences, Kadir Has University, 34083 Istanbul, Turkey
| | - Deniz Eroglu
- Faculty of Engineering and Natural Sciences, Kadir Has University, 34083 Istanbul, Turkey
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17
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Mirzaei S, Anwar MS, Parastesh F, Jafari S, Ghosh D. Synchronization in repulsively coupled oscillators. Phys Rev E 2023; 107:014201. [PMID: 36797861 DOI: 10.1103/physreve.107.014201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2022] [Accepted: 12/01/2022] [Indexed: 01/04/2023]
Abstract
A long-standing expectation is that two repulsively coupled oscillators tend to oscillate in opposite directions. It has been difficult to achieve complete synchrony in coupled identical oscillators with purely repulsive coupling. Here, we introduce a general coupling condition based on the linear matrix of dynamical systems for the emergence of the complete synchronization in pure repulsively coupled oscillators. The proposed coupling profiles (coupling matrices) define a bidirectional cross-coupling link that plays the role of indicator for the onset of complete synchrony between identical oscillators. We illustrate the proposed coupling scheme on several paradigmatic two-coupled chaotic oscillators and validate its effectiveness through the linear stability analysis of the synchronous solution based on the master stability function approach. We further demonstrate that the proposed general condition for the selection of coupling profiles to achieve synchronization even works perfectly for a large ensemble of oscillators.
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Affiliation(s)
- Simin Mirzaei
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 1591634311, Iran
| | - Md Sayeed Anwar
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Fatemeh Parastesh
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 1591634311, Iran
| | - Sajad Jafari
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 1591634311, Iran.,Health Technology Research Institute, Amirkabir University of Technology (Tehran Polytechnic), 1591634311, Iran
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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Nezhad Hajian D, Parthasarathy S, Parastesh F, Rajagopal K, Jafari S. Dominant Attractor in Coupled Non-Identical Chaotic Systems. ENTROPY (BASEL, SWITZERLAND) 2022; 24:1807. [PMID: 36554212 PMCID: PMC9778076 DOI: 10.3390/e24121807] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/07/2022] [Revised: 11/30/2022] [Accepted: 12/09/2022] [Indexed: 06/17/2023]
Abstract
The dynamical interplay of coupled non-identical chaotic oscillators gives rise to diverse scenarios. The incoherent dynamics of these oscillators lead to the structural impairment of attractors in phase space. This paper investigates the couplings of Lorenz-Rössler, Lorenz-HR, and Rössler-HR to identify the dominant attractor. By dominant attractor, we mean the attractor that is less changed by coupling. For comparison and similarity detection, a cost function based on the return map of the coupled systems is used. The possible effects of frequency and amplitude differences between the systems on the results are also examined. Finally, the inherent chaotic characteristic of systems is compared by computing the largest Lyapunov exponent. The results suggest that in each coupling case, the attractor with the greater largest Lyapunov exponent is dominant.
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Affiliation(s)
- Dorsa Nezhad Hajian
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 159163-4311, Iran
| | - Sriram Parthasarathy
- Centre for Computational Modelling, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
| | - Fatemeh Parastesh
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 159163-4311, Iran
| | - Karthikeyan Rajagopal
- Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
| | - Sajad Jafari
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 159163-4311, Iran
- Health Technology Research Institute, Amirkabir University of Technology (Tehran Polytechnic), Tehran 159163-4311, Iran
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19
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Guo H, Zhou J, Zhu S. The impact of inner-coupling and time delay on synchronization: From single-layer network to hypernetwork. CHAOS (WOODBURY, N.Y.) 2022; 32:113135. [PMID: 36456352 DOI: 10.1063/5.0091626] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2022] [Accepted: 10/26/2022] [Indexed: 06/17/2023]
Abstract
Though synchronization of complex dynamical systems has been widely studied in the past few decades, few studies pay attention to the impact of network parameters on synchronization in hypernetworks. In this paper, we focus on a specific hypernetwork model consisting of coupled Rössler oscillators and investigate the impact of inner-coupling and time delay on the synchronized region (SR). For the sake of simplicity, the inner-coupling matrix is chosen from three typical forms, which result in classical bounded, unbounded, and empty SR in a single-layer network, respectively. The impact of inner-couplings or time delays on unbounded SR is the most interesting one among the three types of SR. Once the SR of one subnetwork is unbounded, the SR of the whole hypernetwork is also unbounded with a different inner-coupling matrix. In a hypernetwork with unbounded SR, the time delays change not only the size but also the type of SR. In a hypernetwork with bounded or empty SR, the time delays have almost no effect on the type of SR.
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Affiliation(s)
- Heng Guo
- School of Mathematics and Statistics, Wuhan University, Hubei 430072, China
| | - Jin Zhou
- School of Mathematics and Statistics, Wuhan University, Hubei 430072, China
| | - Shuaibing Zhu
- MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Hunan 410081, China
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20
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Wang Y, Zhang D, Wang L, Li Q, Cao H, Wang X. Cluster synchronization induced by manifold deformation. CHAOS (WOODBURY, N.Y.) 2022; 32:093139. [PMID: 36182364 DOI: 10.1063/5.0107866] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2022] [Accepted: 08/29/2022] [Indexed: 06/16/2023]
Abstract
Pinning control of cluster synchronization in a globally connected network of chaotic oscillators is studied. It is found in simulations that when the pinning strength exceeds a critical value, the oscillators are synchronized into two different clusters, one formed by the pinned oscillators and the other one formed by the unpinned oscillators. The numerical results are analyzed by the generalized method of master stability function (MSF), in which it is shown that whereas the method is able to predict the synchronization behaviors of the pinned oscillators, it fails to predict the synchronization behaviors of the unpinned oscillators. By checking the trajectories of the oscillators in the phase space, it is found that the failure is attributed to the deformed synchronization manifold of the unpinned oscillators, which is clearly deviated from that of isolated oscillator under strong pinnings. A similar phenomenon is also observed in the pinning control of cluster synchronization in a complex network of symmetric structures and in the self-organized cluster synchronization of networked neural oscillators. The findings are important complements to the generalized MSF method and provide an alternative approach to the manipulation of synchronization behaviors in complex network systems.
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Affiliation(s)
- Ya Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Dapeng Zhang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Liang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Qing Li
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Hui Cao
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
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21
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Sendiña-Nadal I, Letellier C. Observability analysis and state reconstruction for networks of nonlinear systems. CHAOS (WOODBURY, N.Y.) 2022; 32:083109. [PMID: 36049910 DOI: 10.1063/5.0090239] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2022] [Accepted: 07/08/2022] [Indexed: 06/15/2023]
Abstract
We address the problem of retrieving the full state of a network of Rössler systems from the knowledge of the actual state of a limited set of nodes. The selection of nodes where sensors are placed is carried out in a hierarchical way through a procedure based on graphical and symbolic observability approaches applied to pairs of coupled dynamical systems. By using a map directly obtained from governing equations, we design a nonlinear network reconstructor that is able to unfold the state of non-measured nodes with working accuracy. For sparse networks, the number of sensor scales with half the network size and node reconstruction errors are lower in networks with heterogeneous degree distributions. The method performs well even in the presence of parameter mismatch and non-coherent dynamics and for dynamical systems with completely different algebraic structures like the Hindmarsch-Rose; therefore, we expect it to be useful for designing robust network control laws.
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Affiliation(s)
- Irene Sendiña-Nadal
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
| | - Christophe Letellier
- Rouen Normandie Université-CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France
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22
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Bhandary S, Biswas D, Banerjee T, Dutta PS. Effects of time-varying habitat connectivity on metacommunity persistence. Phys Rev E 2022; 106:014309. [PMID: 35974633 DOI: 10.1103/physreve.106.014309] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2022] [Accepted: 07/05/2022] [Indexed: 06/15/2023]
Abstract
Network structure or connectivity patterns are critical in determining collective dynamics among interacting species in ecosystems. Conventional research on species persistence in spatial populations has focused on static network structure, though most real network structures change in time, forming time-varying networks. This raises the question, in metacommunities, how does the pattern of synchrony vary with temporal evolution in the network structure. The synchronous dynamics among species are known to reduce metacommunity persistence. Here we consider a time-varying metacommunity small-world network consisting of a chaotic three-species food chain oscillator in each patch or node. The rate of change in the network connectivity is determined by the natural frequency or its subharmonics of the constituent oscillator to allow sufficient time for the evolution of species in between successive rewirings. We find that over a range of coupling strengths and rewiring periods, even higher rewiring probabilities drive a network from asynchrony towards synchrony. Moreover, in networks with a small rewiring period, an increase in average degree (more connected networks) pushes the asynchronous dynamics to synchrony. On the other hand, in networks with a low average degree, a higher rewiring period drives the synchronous dynamics to asynchrony resulting in increased species persistence. Our results also follow the calculation of synchronization time and are robust across other ecosystem models. Overall, our study opens the possibility of developing temporal connectivity strategies to increase species persistence in ecological networks.
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Affiliation(s)
- Subhendu Bhandary
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India
| | - Debabrata Biswas
- Department of Physics, Bankura University, Bankura 722155, West Bengal, India
| | - Tanmoy Banerjee
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan 713104, West Bengal, India
| | - Partha Sharathi Dutta
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India
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23
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Parastesh F, Rajagopal K, Jafari S, Perc M, Schöll E. Blinking coupling enhances network synchronization. Phys Rev E 2022; 105:054304. [PMID: 35706266 DOI: 10.1103/physreve.105.054304] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2021] [Accepted: 04/13/2022] [Indexed: 06/15/2023]
Abstract
This paper studies the synchronization of a network with linear diffusive coupling, which blinks between the variables periodically. The synchronization of the blinking network in the case of sufficiently fast blinking is analyzed by showing that the stability of the synchronous solution depends only on the averaged coupling and not on the instantaneous coupling. To illustrate the effect of the blinking period on the network synchronization, the Hindmarsh-Rose model is used as the dynamics of nodes. The synchronization is investigated by considering constant single-variable coupling, averaged coupling, and blinking coupling through a linear stability analysis. It is observed that by decreasing the blinking period, the required coupling strength for synchrony is reduced. It equals that of the averaged coupling model times the number of variables. However, in the averaged coupling, all variables participate in the coupling, while in the blinking model only one variable is coupled at any time. Therefore, the blinking coupling leads to an enhanced synchronization in comparison with the single-variable coupling. Numerical simulations of the average synchronization error of the network confirm the results obtained from the linear stability analysis.
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Affiliation(s)
- Fatemeh Parastesh
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Iran
| | | | - Sajad Jafari
- Department of Biomedical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Iran
- Health Technology Research Institute, Amirkabir University of Technology (Tehran Polytechnic), Iran
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška Cesta 160, 2000 Maribor, Slovenia
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404332, Taiwan
- Complexity Science Hub Vienna, Josefstädterstraße 39, 1080 Vienna, Austria
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität, D-10115 Berlin, Germany
- Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, D-14473 Potsdam, Germany
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24
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Long YS, Zhai ZM, Tang M, Lai YC. Metamorphoses and explosively remote synchronization in dynamical networks. CHAOS (WOODBURY, N.Y.) 2022; 32:043110. [PMID: 35489847 DOI: 10.1063/5.0088989] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2022] [Accepted: 03/14/2022] [Indexed: 06/14/2023]
Abstract
We uncover a phenomenon in coupled nonlinear networks with a symmetry: as a bifurcation parameter changes through a critical value, synchronization among a subset of nodes can deteriorate abruptly, and, simultaneously, perfect synchronization emerges suddenly among a different subset of nodes that are not directly connected. This is a synchronization metamorphosis leading to an explosive transition to remote synchronization. The finding demonstrates that an explosive onset of synchrony and remote synchronization, two phenomena that have been studied separately, can arise in the same system due to symmetry, providing another proof that the interplay between nonlinear dynamics and symmetry can lead to a surprising phenomenon in physical systems.
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Affiliation(s)
- Yong-Shang Long
- State Key Laboratory of Precision Spectroscopy and School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
| | - Zheng-Meng Zhai
- State Key Laboratory of Precision Spectroscopy and School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
| | - Ming Tang
- State Key Laboratory of Precision Spectroscopy and School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
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25
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Nazerian A, Panahi S, Leifer I, Phillips D, Makse HA, Sorrentino F. Matryoshka and disjoint cluster synchronization of networks. CHAOS (WOODBURY, N.Y.) 2022; 32:041101. [PMID: 35489844 PMCID: PMC8983070 DOI: 10.1063/5.0076412] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2021] [Accepted: 03/14/2022] [Indexed: 06/14/2023]
Abstract
The main motivation for this paper is to characterize network synchronizability for the case of cluster synchronization (CS), in an analogous fashion to Barahona and Pecora [Phys. Rev. Lett. 89, 054101 (2002)] for the case of complete synchronization. We find this problem to be substantially more complex than the original one. We distinguish between the two cases of networks with intertwined clusters and no intertwined clusters and between the two cases that the master stability function is negative either in a bounded range or in an unbounded range of its argument. Our proposed definition of cluster synchronizability is based on the synchronizability of each individual cluster within a network. We then attempt to generalize this definition to the entire network. For CS, the synchronous solution for each cluster may be stable, independent of the stability of the other clusters, which results in possibly different ranges in which each cluster synchronizes (isolated CS). For each pair of clusters, we distinguish between three different cases: Matryoshka cluster synchronization (when the range of the stability of the synchronous solution for one cluster is included in that of the other cluster), partially disjoint cluster synchronization (when the ranges of stability of the synchronous solutions partially overlap), and complete disjoint cluster synchronization (when the ranges of stability of the synchronous solutions do not overlap).
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Affiliation(s)
- Amirhossein Nazerian
- Department of Mechanical Engineering, University of New Mexico, Albuquerque, New Mexico 87131, USA
| | - Shirin Panahi
- Department of Mechanical Engineering, University of New Mexico, Albuquerque, New Mexico 87131, USA
| | - Ian Leifer
- Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA
| | - David Phillips
- Department of Mathematics, United States Naval Academy, Annapolis, Maryland 21401, USA
| | - Hernán A. Makse
- Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA
| | - Francesco Sorrentino
- Department of Mechanical Engineering, University of New Mexico, Albuquerque, New Mexico 87131, USA
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Abstract
A chaotic dynamics is typically characterized by the emergence of strange attractors with their fractal or multifractal structure. On the other hand, chaotic synchronization is a unique emergent self-organization phenomenon in nature. Classically, synchronization was characterized in terms of macroscopic parameters, such as the spectrum of Lyapunov exponents. Recently, however, we attempted a microscopic description of synchronization, called topological synchronization, and showed that chaotic synchronization is, in fact, a continuous process that starts in low-density areas of the attractor. Here we analyze the relation between the two emergent phenomena by shifting the descriptive level of topological synchronization to account for the multifractal nature of the visited attractors. Namely, we measure the generalized dimension of the system and monitor how it changes while increasing the coupling strength. We show that during the gradual process of topological adjustment in phase space, the multifractal structures of each strange attractor of the two coupled oscillators continuously converge, taking a similar form, until complete topological synchronization ensues. According to our results, chaotic synchronization has a specific trait in various systems, from continuous systems and discrete maps to high dimensional systems: synchronization initiates from the sparse areas of the attractor, and it creates what we termed as the ‘zipper effect’, a distinctive pattern in the multifractal structure of the system that reveals the microscopic buildup of the synchronization process. Topological synchronization offers, therefore, a more detailed microscopic description of chaotic synchronization and reveals new information about the process even in cases of high mismatch parameters.
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27
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Diggans CT, Fish J, AlMomani AAR, Bollt EM. The essential synchronization backbone problem. CHAOS (WOODBURY, N.Y.) 2021; 31:113142. [PMID: 34881620 DOI: 10.1063/5.0065077] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2021] [Accepted: 11/04/2021] [Indexed: 06/13/2023]
Abstract
Network optimization strategies for the process of synchronization have generally focused on the re-wiring or re-weighting of links in order to (1) expand the range of coupling strengths that achieve synchronization, (2) expand the basin of attraction for the synchronization manifold, or (3) lower the average time to synchronization. A new optimization goal is proposed in seeking the minimum subset of the edge set of the original network that enables the same essential ability to synchronize in that the synchronization manifolds have conjugate stability. We call this type of minimal spanning subgraph an essential synchronization backbone of the original system, and we present two algorithms: one is a strategy for an exhaustive search for a true solution, while the other is a method of approximation for this combinatorial problem. The solution spaces that result from different choices of dynamical systems and coupling schemes vary with the level of a hierarchical structure present and also the number of interwoven central cycles. Applications can include the important problem in civil engineering of power grid hardening, where new link creation may be costly, and the defense of certain key links to the functional process may be prioritized.
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Affiliation(s)
- C Tyler Diggans
- Air Force Research Laboratory Information Directorate, Rome, New York 13441, USA
| | - Jeremie Fish
- Clarkson Center for Complex Systems Science, Potsdam, New York 13669, USA
| | | | - Erik M Bollt
- Clarkson Center for Complex Systems Science, Potsdam, New York 13669, USA
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28
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Letellier C, Sendiña-Nadal I, Minati L, Leyva I. Node differentiation dynamics along the route to synchronization in complex networks. Phys Rev E 2021; 104:014303. [PMID: 34412314 DOI: 10.1103/physreve.104.014303] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2021] [Accepted: 05/11/2021] [Indexed: 11/07/2022]
Abstract
Synchronization has been the subject of intense research during decades mainly focused on determining the structural and dynamical conditions driving a set of interacting units to a coherent state globally stable. However, little attention has been paid to the description of the dynamical development of each individual networked unit in the process towards the synchronization of the whole ensemble. In this paper we show how in a network of identical dynamical systems, nodes belonging to the same degree class, differentiate in the same manner, visiting a sequence of states of diverse complexity along the route to synchronization independently on the global network structure. In particular, we observe, just after interaction starts pulling orbits from the initially uncoupled attractor, a general reduction of the complexity of the dynamics of all units being more pronounced in those with higher connectivity. In the weak-coupling regime, when synchronization starts to build up, there is an increase in the dynamical complexity, whose maximum is achieved, in general, first in the hubs due to their earlier synchronization with the mean field. For very strong coupling, just before complete synchronization, we found a hierarchical dynamical differentiation with lower degree nodes being the ones exhibiting the largest complexity departure. We unveil how this differentiation route holds for several models of nonlinear dynamics, including toroidal chaos and how it depends on the coupling function. This study provides insights to understand better strategies for network identification or to devise effective methods for network inference.
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Affiliation(s)
- Christophe Letellier
- Rouen Normandie University - CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France
| | - Irene Sendiña-Nadal
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain.,Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - Ludovico Minati
- Center for Mind/Brain Sciences (CIMeC), University of Trento, 38123 Trento, Italy.,Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8503, Japan
| | - I Leyva
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain.,Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
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29
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Rajagopal K, Jafari S, Karthikeyan A, Srinivasan A. Effect of magnetic induction on the synchronizability of coupled neuron network. CHAOS (WOODBURY, N.Y.) 2021; 31:083115. [PMID: 34470222 DOI: 10.1063/5.0061406] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2021] [Accepted: 07/28/2021] [Indexed: 06/13/2023]
Abstract
Master stability functions (MSFs) are significant tools to identify the synchronizability of nonlinear dynamical systems. For a network of coupled oscillators to be synchronized, the corresponding MSF should be negative. The study of MSF will normally be discussed considering the coupling factor as a control variable. In our study, we considered various neuron models with electromagnetic flux induction and investigated the MSF's zero-crossing points for various values of the flux coupling coefficient. Our numerical analysis has shown that in all the neuron models we considered, flux coupling has increased the synchronization of the coupled neuron by increasing the number of zero-crossing points of MSFs or by achieving a zero-crossing point for a lesser value of a coupling parameter.
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Affiliation(s)
- Karthikeyan Rajagopal
- Center for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
| | - Sajad Jafari
- Biomedical Engineering Department, Amirkabir University of Technology, Tehran 1591634311, Iran
| | - Anitha Karthikeyan
- Department of Electronics and Communication Engineering, Prathyusha Engineering College, Thiruvallur, Tamil Nadu 602025, India
| | - Ashokkumar Srinivasan
- Center for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
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30
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Khatun AA, Jafri HH, Punetha N. Controlling chimera states in chaotic oscillator ensembles through linear augmentation. Phys Rev E 2021; 103:042202. [PMID: 34005985 DOI: 10.1103/physreve.103.042202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2019] [Accepted: 03/07/2021] [Indexed: 11/07/2022]
Abstract
In this work, we show how "chimera states," namely, the dynamical situation when synchronized and desynchronized domains coexist in an oscillator ensemble, can be controlled through a linear augmentation (LA) technique. Specifically, in the networks of coupled chaotic oscillators, we obtain chimera states through induced multistability and demonstrate how LA can be used to control the size and spatial location of the incoherent and coherent populations in the ensemble. We examine basins of attraction of the system to analyze the effects of LA on its multistable behavior and thus on chimera states. Stability of the synchronized dynamics is analyzed through a master stability function. We find that these results are independent of a system's initial conditions and the strategy is applicable to the networks of globally, locally as well as nonlocally coupled oscillators. Our results suggest that LA control can be an effective method to control chimera states and to realize a desired collective dynamics in such ensembles.
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Affiliation(s)
- Anjuman Ara Khatun
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
| | - Haider Hasan Jafri
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
| | - Nirmal Punetha
- Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
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31
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Gambuzza LV, Di Patti F, Gallo L, Lepri S, Romance M, Criado R, Frasca M, Latora V, Boccaletti S. Stability of synchronization in simplicial complexes. Nat Commun 2021; 12:1255. [PMID: 33623044 PMCID: PMC7902853 DOI: 10.1038/s41467-021-21486-9] [Citation(s) in RCA: 50] [Impact Index Per Article: 16.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2020] [Accepted: 01/12/2021] [Indexed: 01/31/2023] Open
Abstract
Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.
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Affiliation(s)
- L V Gambuzza
- Department of Electrical, Electronics and Computer Science Engineering, University of Catania, Catania, Italy
| | - F Di Patti
- CNR-Institute of Complex Systems, Florence, Italy
| | - L Gallo
- Department of Physics and Astronomy, University of Catania, Catania, Italy
- INFN Sezione di Catania, Catania, Italy
| | - S Lepri
- CNR-Institute of Complex Systems, Florence, Italy
| | - M Romance
- Department of Applied Math. and Data, Complex Networks and Cybersecurity Research Institute, University Rey Juan Carlos, Madrid, Spain
| | - R Criado
- Department of Applied Math. and Data, Complex Networks and Cybersecurity Research Institute, University Rey Juan Carlos, Madrid, Spain
| | - M Frasca
- Department of Electrical, Electronics and Computer Science Engineering, University of Catania, Catania, Italy.
- Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti", Consiglio Nazionale delle Ricerche (IASI-CNR), Roma, Italy.
| | - V Latora
- Department of Physics and Astronomy, University of Catania, Catania, Italy.
- INFN Sezione di Catania, Catania, Italy.
- School of Mathematical Sciences, Queen Mary University of London, London, UK.
- The Alan Turing Institute, The British Library, London, UK.
| | - S Boccaletti
- CNR-Institute of Complex Systems, Florence, Italy.
- Unmanned Systems Research Institute, Northwestern Polytechnical University, Xi'an, China.
- Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russian Federation.
- Universidad Rey Juan Carlos, Móstoles, Madrid, Spain.
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32
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Guo Y, Zhang H, Wang L, Fan H, Xiao J, Wang X. Transfer learning of chaotic systems. CHAOS (WOODBURY, N.Y.) 2021; 31:011104. [PMID: 33754764 DOI: 10.1063/5.0033870] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2020] [Accepted: 12/24/2020] [Indexed: 06/12/2023]
Abstract
Can a neural network trained by the time series of system A be used to predict the evolution of system B? This problem, knowing as transfer learning in a broad sense, is of great importance in machine learning and data mining yet has not been addressed for chaotic systems. Here, we investigate transfer learning of chaotic systems from the perspective of synchronization-based state inference, in which a reservoir computer trained by chaotic system A is used to infer the unmeasured variables of chaotic system B, while A is different from B in either parameter or dynamics. It is found that if systems A and B are different in parameter, the reservoir computer can be well synchronized to system B. However, if systems A and B are different in dynamics, the reservoir computer fails to synchronize with system B in general. Knowledge transfer along a chain of coupled reservoir computers is also studied, and it is found that, although the reservoir computers are trained by different systems, the unmeasured variables of the driving system can be successfully inferred by the remote reservoir computer. Finally, by an experiment of chaotic pendulum, we demonstrate that the knowledge learned from the modeling system can be transferred and used to predict the evolution of the experimental system.
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Affiliation(s)
- Yali Guo
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Han Zhang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Liang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Huawei Fan
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Jinghua Xiao
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
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33
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Njougouo T, Camargo V, Louodop P, Fagundes Ferreira F, Talla PK, Cerdeira HA. Dynamics of multilayer networks with amplification. CHAOS (WOODBURY, N.Y.) 2020; 30:123136. [PMID: 33380025 DOI: 10.1063/5.0025529] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2020] [Accepted: 12/01/2020] [Indexed: 06/12/2023]
Abstract
We study the dynamics of a multilayer network of chaotic oscillators subject to amplification. Previous studies have proven that multilayer networks present phenomena such as synchronization, cluster, and chimera states. Here, we consider a network with two layers of Rössler chaotic oscillators as well as applications to multilayer networks of the chaotic jerk and Liénard oscillators. Intra-layer coupling is considered to be all to all in the case of Rössler oscillators, a ring for jerk oscillators and global mean field coupling in the case of Liénard, inter-layer coupling is unidirectional in all these three cases. The second layer has an amplification coefficient. An in-depth study on the case of a network of Rössler oscillators using a master stability function and order parameter leads to several phenomena such as complete synchronization, generalized, cluster, and phase synchronization with amplification. For the case of Rössler oscillators, we note that there are also certain values of coupling parameters and amplification where the synchronization does not exist or the synchronization can exist but without amplification. Using other systems with different topologies, we obtain some interesting results such as chimera state with amplification, cluster state with amplification, and complete synchronization with amplification.
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Affiliation(s)
- Thierry Njougouo
- Research Unit Condensed Matter, Electronics and Signal Processing, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - Victor Camargo
- Center for Interdisciplinary Research on Complex Systems, University of Sao Paulo, Av. Arlindo Bettio 1000, 03828-000 São Paulo, Brazil
| | - Patrick Louodop
- Research Unit Condensed Matter, Electronics and Signal Processing, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - Fernando Fagundes Ferreira
- Center for Interdisciplinary Research on Complex Systems, University of Sao Paulo, Av. Arlindo Bettio 1000, 03828-000 São Paulo, Brazil
| | - Pierre K Talla
- L2MSP, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - Hilda A Cerdeira
- São Paulo State University (UNESP), Instituto de Física Teórica, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil
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34
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Vera-Ávila VP, Sevilla-Escoboza JR, Leyva I. Complex networks exhibit intermittent synchronization. CHAOS (WOODBURY, N.Y.) 2020; 30:103119. [PMID: 33138450 DOI: 10.1063/5.0020419] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2020] [Accepted: 09/28/2020] [Indexed: 06/11/2023]
Abstract
The path toward the synchronization of an ensemble of dynamical units goes through a series of transitions determined by the dynamics and the structure of the connections network. In some systems on the verge of complete synchronization, intermittent synchronization, a time-dependent state where full synchronization alternates with non-synchronized periods, has been observed. This phenomenon has been recently considered to have functional relevance in neuronal ensembles and other networked biological systems close to criticality. We characterize the intermittent state as a function of the network topology to show that the different structures can encourage or inhibit the appearance of early signs of intermittency. In particular, we study the local intermittency and show how the nodes incorporate to intermittency in hierarchical order, which can provide information about the node topological role even when the structure is unknown.
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Affiliation(s)
- V P Vera-Ávila
- Centro Universitario de los Lagos, Universidad de Guadalajara, Jalisco 47460, Mexico
| | - J R Sevilla-Escoboza
- Centro Universitario de los Lagos, Universidad de Guadalajara, Jalisco 47460, Mexico
| | - I Leyva
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain and Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Madrid, Spain
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35
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Wu X, Li Q, Liu C, Liu J, Xie C. Synchronization in duplex networks of coupled Rössler oscillators with different inner-coupling matrices. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2019.10.011] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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36
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Arena P, Buscarino A, Fortuna L, Patanè L. Lyapunov approach to synchronization of chaotic systems with vanishing nonlinear perturbations: From static to dynamic couplings. Phys Rev E 2020; 102:012211. [PMID: 32794950 DOI: 10.1103/physreve.102.012211] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2020] [Accepted: 06/26/2020] [Indexed: 11/07/2022]
Abstract
Synchronization of chaotic dynamics can be pursued by means of different coupling strategies. Definitely, master-slave coupling represents one of the most adopted solutions, even if it presents some limitations due to the coupling term's selection strategy. In this paper, we investigate the role of different structures of coupling terms on the synchronization properties of master-slave chaotic system configurations. Here, Lyapunov theory for linear systems with nonlinear vanishing perturbations is exploited. The obtained results allow to determine the capability of a static, dynamic, or mixed coupling connection in stabilizing the synchronization manifold, using linear techniques based on the root locus. This knowledge allows to design the coupling structure considering also the synchronization error transient features, which are, here, shown to improve in the presence of higher-order dynamic couplings. A number of cases of study, involving classical chaotic nonlinear systems, show the efficacy and simplicity of the application of the strategy proposed.
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Affiliation(s)
- Paolo Arena
- DIEEI, University of Catania, Viale A. Doria 6, Catania 95125, Italy and CNR-IASI, Istituto di Analisi del Sistemi e Informatica "A. Ruberti," 00185 Rome, Italy
| | - Arturo Buscarino
- DIEEI, University of Catania, Viale A. Doria 6, Catania 95125, Italy and CNR-IASI, Istituto di Analisi del Sistemi e Informatica "A. Ruberti," 00185 Rome, Italy
| | - Luigi Fortuna
- DIEEI, University of Catania, Viale A. Doria 6, Catania 95125, Italy and CNR-IASI, Istituto di Analisi del Sistemi e Informatica "A. Ruberti," 00185 Rome, Italy
| | - Luca Patanè
- Dipartimento di Ingegneria Università degli Studi di Messina, Contrada di Dio, 98166 Messina, Italy
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37
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Tang E, Ju H, Baum GL, Roalf DR, Satterthwaite TD, Pasqualetti F, Bassett DS. Control of brain network dynamics across diverse scales of space and time. Phys Rev E 2020; 101:062301. [PMID: 32688528 PMCID: PMC8728948 DOI: 10.1103/physreve.101.062301] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2019] [Accepted: 03/12/2020] [Indexed: 12/30/2022]
Abstract
The human brain is composed of distinct regions that are each associated with particular functions and distinct propensities for the control of neural dynamics. However, the relation between these functions and control profiles is poorly understood, as is the variation in this relation across diverse scales of space and time. Here we probe the relation between control and dynamics in brain networks constructed from diffusion tensor imaging data in a large community sample of young adults. Specifically, we probe the control properties of each brain region and investigate their relationship with dynamics across various spatial scales using the Laplacian eigenspectrum. In addition, through analysis of regional modal controllability and partitioning of modes, we determine whether the associated dynamics are fast or slow, as well as whether they are alternating or monotone. We find that brain regions that facilitate the control of energetically easy transitions are associated with activity on short length scales and slow timescales. Conversely, brain regions that facilitate control of difficult transitions are associated with activity on long length scales and fast timescales. Built on linear dynamical models, our results offer parsimonious explanations for the activity propagation and network control profiles supported by regions of differing neuroanatomical structure.
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Affiliation(s)
- Evelyn Tang
- Department of Bioengineering, School of Engineering & Applied Science, University of Pennsylvania, Pennsylvania 19104, USA
- Max Planck Institute for Dynamics and Self-Organization, Göttingen 37079, Germany
| | - Harang Ju
- Department of Bioengineering, School of Engineering & Applied Science, University of Pennsylvania, Pennsylvania 19104, USA
- Neuroscience Graduate Program, Perelman School of Medicine, University of Pennsylvania, Pennsylvania 19104, USA
| | - Graham L Baum
- Department of Bioengineering, School of Engineering & Applied Science, University of Pennsylvania, Pennsylvania 19104, USA
- Neuroscience Graduate Program, Perelman School of Medicine, University of Pennsylvania, Pennsylvania 19104, USA
| | - David R Roalf
- Department of Psychiatry, Perelman School of Medicine, University of Pennsylvania, Pennsylvania 19104, USA
| | - Theodore D Satterthwaite
- Department of Psychiatry, Perelman School of Medicine, University of Pennsylvania, Pennsylvania 19104, USA
| | - Fabio Pasqualetti
- Department of Mechanical Engineering, University of California, Riverside, Riverside, California 92521, USA
| | - Danielle S Bassett
- Department of Bioengineering, School of Engineering & Applied Science, University of Pennsylvania, Pennsylvania 19104, USA
- Department of Psychiatry, Perelman School of Medicine, University of Pennsylvania, Pennsylvania 19104, USA
- Department of Physics & Astronomy, College of Arts & Sciences, University of Pennsylvania, Pennsylvania 19104, USA
- Department of Electrical & Systems Engineering, School of Engineering & Applied Science, University of Pennsylvania, Pennsylvania 19104, USA
- Department of Neurology, Perelman School of Medicine, University of Pennsylvania, Pennsylvania 19104, USA
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA
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38
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Yi M, Wang C, Yang K. Discontinuity-induced intermittent synchronization transitions in coupled non-smooth systems. CHAOS (WOODBURY, N.Y.) 2020; 30:033113. [PMID: 32237761 DOI: 10.1063/1.5123299] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2019] [Accepted: 02/14/2020] [Indexed: 06/11/2023]
Abstract
The synchronization transition in coupled non-smooth systems is studied for increasing coupling strength. The average order parameter is calculated to diagnose synchronization of coupled non-smooth systems. It is found that the coupled non-smooth system exhibits an intermittent synchronization transition from the cluster synchronization state to the complete synchronization state, depending on the coupling strength and initial conditions. Detailed numerical analyses reveal that the discontinuity always plays an important role in the synchronization transition of the coupled non-smooth system. In addition, it is found that increasing the coupling strength leads to the coexistence of periodic cluster states. Detailed research illustrates that the periodic clusters consist of two or more coexisting periodic attractors. Their periodic trajectory passes from one region to another region that is divided by discontinuous boundaries in the phase space. The mutual interactions of the local nonlinearity and the spatial coupling ultimately result in a stable periodic trajectory.
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Affiliation(s)
- Ming Yi
- School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
| | - Canjun Wang
- Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721016, China
| | - Keli Yang
- Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721016, China
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39
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Rakshit S, Bera BK, Ghosh D. Invariance and stability conditions of interlayer synchronization manifold. Phys Rev E 2020; 101:012308. [PMID: 32069525 DOI: 10.1103/physreve.101.012308] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/02/2019] [Indexed: 11/07/2022]
Abstract
We investigate interlayer synchronization in a stochastic multiplex hypernetwork which is defined by the two types of connections, one is the intralayer connection in each layer with hypernetwork structure and the other is the interlayer connection between the layers. Here all types of interactions within and between the layers are allowed to vary with a certain rewiring probability. We address the question about the invariance and stability of the interlayer synchronization state in this stochastic multiplex hypernetwork. For the invariance of interlayer synchronization manifold, the adjacency matrices corresponding to each tier in each layer should be equal and the interlayer connection should be either bidirectional or the interlayer coupling function should vanish after achieving the interlayer synchronization state. We analytically derive a necessary-sufficient condition for local stability of the interlayer synchronization state using master stability function approach and a sufficient condition for global stability by constructing a suitable Lyapunov function. Moreover, we analytically derive that intralayer synchronization is unattainable for this network architecture due to stochastic interlayer connections. Remarkably, our derived invariance and stability conditions (both local and global) are valid for any rewiring probabilities, whereas most of the previous stability conditions are only based on a fast switching approximation.
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Affiliation(s)
- Sarbendu Rakshit
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Bidesh K Bera
- Department of Mathematics, Indian Institute of Technology Ropar, Punjab 140001, India.,Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion, 8499000, Israel
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
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40
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Wang Y, Wang L, Fan H, Wang X. Cluster synchronization in networked nonidentical chaotic oscillators. CHAOS (WOODBURY, N.Y.) 2019; 29:093118. [PMID: 31575156 DOI: 10.1063/1.5097242] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2019] [Accepted: 08/26/2019] [Indexed: 06/10/2023]
Abstract
In exploring oscillator synchronization, a general observation is that as the oscillators become nonidentical, e.g., introducing parameter mismatch among the oscillators, the propensity for synchronization will be deteriorated. Yet in realistic systems, parameter mismatch is unavoidable and even worse in some circumstances, the oscillators might follow different types of dynamics. Considering the significance of synchronization to the functioning of many realistic systems, it is natural to ask the following question: Can synchronization be achieved in networked oscillators of clearly different parameters or dynamics? Here, by the model of networked chaotic oscillators, we are able to demonstrate and argue that, despite the presence of parameter mismatch (or different dynamics), stable synchronization can still be achieved on symmetric complex networks. Specifically, we find that when the oscillators are configured on the network in such a way that the symmetric nodes have similar parameters (or follow the same type of dynamics), cluster synchronization can be generated. The stabilities of the cluster synchronization states are analyzed by the method of symmetry-based stability analysis, with the theoretical predictions in good agreement with the numerical results. Our study sheds light on the interplay between symmetry and cluster synchronization in complex networks and give insights into the functionalities of realistic systems where nonidentical nonlinear oscillators are presented and cluster synchronization is crucial.
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Affiliation(s)
- Yafeng Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Liang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Huawei Fan
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
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41
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Buscarino A, Fortuna L, Patanè L. Master-slave synchronization of hyperchaotic systems through a linear dynamic coupling. Phys Rev E 2019; 100:032215. [PMID: 31639891 DOI: 10.1103/physreve.100.032215] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/26/2019] [Indexed: 06/10/2023]
Abstract
The development of synchronization strategies for dynamical systems is an important research activity that can be applied in several different fields from locomotion control of multilimbed structures to secure communication. In the presence of chaotic systems, synchronization is more difficult to accomplish and there are different techniques that can be adopted. In this paper we considered a master-slave topology where the coupling mechanism is realized through a second-order linear dynamical system. This control scheme, recently applied to chaotic systems, is here analyzed in the presence of hyperchaotic dynamics that represent a more challenging scenario. The possibility to reach a complete synchronization and the range of allowable coupling strength is investigated comparing the effects of the dynamical coupling with a standard configuration characterized by a static gain. This methodology is also applied to weighted networks to reach synchronization regimes otherwise not obtainable with a static coupling.
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Affiliation(s)
- Arturo Buscarino
- DIEEI, University of Catania, Viale A. Doria 6 Catania, 95125 Italy
| | - Luigi Fortuna
- DIEEI, University of Catania, Viale A. Doria 6 Catania, 95125 Italy
| | - Luca Patanè
- DIEEI, University of Catania, Viale A. Doria 6 Catania, 95125 Italy
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42
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Zhou S, Guo Y, Liu M, Lai YC, Lin W. Random temporal connections promote network synchronization. Phys Rev E 2019; 100:032302. [PMID: 31639942 DOI: 10.1103/physreve.100.032302] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2018] [Indexed: 06/10/2023]
Abstract
We report a phenomenon of collective dynamics on discrete-time complex networks: a random temporal interaction matrix even of zero or/and small average is able to significantly enhance synchronization with probability one. According to current knowledge, there is no verifiably sufficient criterion for the phenomenon. We use the standard method of synchronization analytics and the theory of stochastic processes to establish a criterion, by which we rigorously and accurately depict how synchronization occurring with probability one is affected by the statistical characteristics of the random temporal connections such as the strength and topology of the connections as well as their probability distributions. We also illustrate the enhancement phenomenon using physical and biological complex dynamical networks.
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Affiliation(s)
- Shijie Zhou
- Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China
- School of Mathematical Science, Fudan University, Shanghai 200433, China
- Shanghai Center of Mathematical Sciences, Shanghai 200433, China
| | - Yao Guo
- Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China
- Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai 200433, China
| | - Maoxing Liu
- Department of Mathematics, North University of China, Taiyuan 030051, China
| | - Ying-Cheng Lai
- School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, Arizona 85287-5706, USA
| | - Wei Lin
- Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China
- School of Mathematical Science, Fudan University, Shanghai 200433, China
- Shanghai Center of Mathematical Sciences, Shanghai 200433, China
- Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai 200433, China
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43
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Karakaya B, Minati L, Gambuzza LV, Frasca M. Fading of remote synchronization in tree networks of Stuart-Landau oscillators. Phys Rev E 2019; 99:052301. [PMID: 31212500 DOI: 10.1103/physreve.99.052301] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2019] [Indexed: 11/07/2022]
Abstract
Remote synchronization (RS) is characterized by the appearance of phase coherence between oscillators that do not directly interact through a structural link in a network but exclusively through other units that are not synchronized or more weakly synchronized with them. This form of phase synchronization was observed initially in starlike motifs and later in random networks. In this paper, we report on an experimental setup for the analysis of RS in networks of Stuart-Landau oscillators and in particular investigate the behavior of tree structures focusing on the path to synchronization, that is, on the analysis of how synchronization emerges as the coupling strength increases from zero. We find that RS occurs in a region wherein further increases of the coupling strength lead to a direct transition to global synchronization but may also be observed in a second region, corresponding to lower coupling values, wherein it first emerges and then disappears, hallmarking a scenario that we denote as fading of remote synchronization. We show that this result is related to the behavior of pairs of remotely synchronized nodes observed in networks with more general topologies. Experiments are corroborated by numerical simulations confirming the major findings and providing further characterization of the phenomenon. We demonstrate that the distribution of natural oscillation frequencies and the parameter uncertainty in the links both play a fundamental role in shaping the behaviors observed.
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Affiliation(s)
- Baris Karakaya
- Faculty of Engineering, Department of Electrical, Electronics Engineering, Firat University, 23119 Elazig, Turkey
| | - Ludovico Minati
- World Research Hub Initiative-Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8503, Japan; Complex Systems Theory Department, Institute of Nuclear Physics-Polish Academy of Sciences (IFJ-PAN), 31-342 Kraków, Poland; and Center for Mind/Brain Sciences (CIMeC), University of Trento, 38123 Trento, Italy
| | - Lucia Valentina Gambuzza
- Dipartimento Ingegneria Elettrica Elettronica e Informatica, Università degli Studi di Catania, 95029 Catania, Italy
| | - Mattia Frasca
- Dipartimento Ingegneria Elettrica Elettronica e Informatica, Università degli Studi di Catania, 95029 Catania, Italy.,CNR-IASI, Italian National Research Council-Institute for Systems Analysis and Computer Science "A. Ruberti," 00185 Rome, Italy
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44
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Fan H, Wang Y, Yang K, Wang X. Enhancing network synchronizability by strengthening a single node. Phys Rev E 2019; 99:042305. [PMID: 31108592 DOI: 10.1103/physreve.99.042305] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2018] [Indexed: 11/07/2022]
Abstract
In improving the stability of complex dynamical systems, an outstanding problem is how to achieve the desired performance at a low cost. For engineering and biological complex systems whose performance and functionality rely on the synchronous motion of their units, an important question related to the performance-cost-balance problem is how to improve efficiently the system synchronizability when a small amount of additional coupling resource is available. Here, employing a complex network of coupled chaotic oscillators as the model, we address this question by introducing a small amount of coupling intensity to only a single oscillator and investigate how the improvement of the network synchronizability is dependent on the location of the target oscillator. Theoretical analysis shows that, to achieve the maximum network synchronizability, the target oscillator to be strengthened should be chosen according to the eigenvector of the most unstable mode. Based on the theoretical finding, we further propose a single-node-based scheme for improving synchronization: the eigenvector-centrality-based strengthening scheme. We describe in detail how to apply this scheme under different synchronization scenarios and justify its efficiency in various network models by numerical simulations. The performance of the new scheme is compared with the conventional ones based on betweenness, closeness, and degree centralities, and it is shown that the new scheme has a clear advantage over the conventional ones. Furthermore, by a brute-force search of the target oscillator over the network, it is verified numerically that the oscillator identified by the new scheme indeed gives the best synchronization performance.
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Affiliation(s)
- Huawei Fan
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Yafeng Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Kai Yang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
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45
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Papo D, Buldú JM. Brain synchronizability, a false friend. Neuroimage 2019; 196:195-199. [PMID: 30986500 DOI: 10.1016/j.neuroimage.2019.04.029] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2018] [Revised: 03/28/2019] [Accepted: 04/08/2019] [Indexed: 01/20/2023] Open
Abstract
Synchronization plays a fundamental role in healthy cognitive and motor function. However, how synchronization depends on the interplay between local dynamics, coupling and topology and how prone to synchronization a network is, given its topological organization, are still poorly understood issues. To investigate the synchronizability of both anatomical and functional brain networks various studies resorted to the Master Stability Function (MSF) formalism, an elegant tool which allows analysing the stability of synchronous states in a dynamical system consisting of many coupled oscillators. Here, we argue that brain dynamics does not fulfil the formal criteria under which synchronizability is usually quantified and, perhaps more importantly, this measure refers to a global dynamical condition that never holds in the brain (not even in the most pathological conditions), and therefore no neurophysiological conclusions should be drawn based on it. We discuss the meaning of synchronizability and its applicability to neuroscience and propose alternative ways to quantify brain networks synchronization.
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Affiliation(s)
- D Papo
- SCALab UMR CNRS 9193, Université de Lille, Villeneuve d'Ascq, France.
| | - J M Buldú
- Laboratory of Biological Networks, Center for Biomedical Technology (UPM), 28223, Pozuelo de Alarcón, Madrid, Spain; Complex Systems Group & G.I.S.C., Universidad Rey Juan Carlos, 28933, Móstoles, Madrid, Spain
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Hart JD, Zhang Y, Roy R, Motter AE. Topological Control of Synchronization Patterns: Trading Symmetry for Stability. PHYSICAL REVIEW LETTERS 2019; 122:058301. [PMID: 30822003 DOI: 10.1103/physrevlett.122.058301] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2018] [Revised: 01/08/2019] [Indexed: 06/09/2023]
Abstract
Symmetries are ubiquitous in network systems and have profound impacts on the observable dynamics. At the most fundamental level, many synchronization patterns are induced by underlying network symmetry, and a high degree of symmetry is believed to enhance the stability of identical synchronization. Yet, here we show that the synchronizability of almost any symmetry cluster in a network of identical nodes can be enhanced precisely by breaking its structural symmetry. This counterintuitive effect holds for generic node dynamics and arbitrary network structure and is, moreover, robust against noise and imperfections typical of real systems, which we demonstrate by implementing a state-of-the-art optoelectronic experiment. These results lead to new possibilities for the topological control of synchronization patterns, which we substantiate by presenting an algorithm that optimizes the structure of individual clusters under various constraints.
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Affiliation(s)
- Joseph D Hart
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
- Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Yuanzhao Zhang
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
| | - Rajarshi Roy
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
- Department of Physics, University of Maryland, College Park, Maryland 20742, USA
- Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
| | - Adilson E Motter
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
- Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA
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Toker D, Sommer FT. Information integration in large brain networks. PLoS Comput Biol 2019; 15:e1006807. [PMID: 30730907 PMCID: PMC6382174 DOI: 10.1371/journal.pcbi.1006807] [Citation(s) in RCA: 30] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2018] [Revised: 02/20/2019] [Accepted: 01/20/2019] [Indexed: 11/30/2022] Open
Abstract
An outstanding problem in neuroscience is to understand how information is integrated across the many modules of the brain. While classic information-theoretic measures have transformed our understanding of feedforward information processing in the brain's sensory periphery, comparable measures for information flow in the massively recurrent networks of the rest of the brain have been lacking. To address this, recent work in information theory has produced a sound measure of network-wide "integrated information", which can be estimated from time-series data. But, a computational hurdle has stymied attempts to measure large-scale information integration in real brains. Specifically, the measurement of integrated information involves a combinatorial search for the informational "weakest link" of a network, a process whose computation time explodes super-exponentially with network size. Here, we show that spectral clustering, applied on the correlation matrix of time-series data, provides an approximate but robust solution to the search for the informational weakest link of large networks. This reduces the computation time for integrated information in large systems from longer than the lifespan of the universe to just minutes. We evaluate this solution in brain-like systems of coupled oscillators as well as in high-density electrocortigraphy data from two macaque monkeys, and show that the informational "weakest link" of the monkey cortex splits posterior sensory areas from anterior association areas. Finally, we use our solution to provide evidence in support of the long-standing hypothesis that information integration is maximized by networks with a high global efficiency, and that modular network structures promote the segregation of information.
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Affiliation(s)
- Daniel Toker
- Helen Wills Neuroscience Institute, University of California Berkeley, Berkeley, California, United States of America
| | - Friedrich T. Sommer
- Helen Wills Neuroscience Institute, University of California Berkeley, Berkeley, California, United States of America
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48
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Ge P, Cao H. Synchronization of Rulkov neuron networks coupled by excitatory and inhibitory chemical synapses. CHAOS (WOODBURY, N.Y.) 2019; 29:023129. [PMID: 30823734 DOI: 10.1063/1.5053908] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2018] [Accepted: 02/01/2019] [Indexed: 06/09/2023]
Abstract
This paper takes into account a neuron network model in which the excitatory and the inhibitory Rulkov neurons interact each other through excitatory and inhibitory chemical coupling, respectively. Firstly, for two or more identical or non-identical Rulkov neurons, the existence conditions of the synchronization manifold of the fixed points are investigated, which have received less attention over the past decades. Secondly, the master stability equation of the arbitrarily connected neuron network under the existence conditions of the synchronization manifold is discussed. Thirdly, taking three identical Rulkov neurons as an example, some new results are presented: (1) topological structures that can make the synchronization manifold exist are given, (2) the stability of synchronization when different parameters change is discussed, and (3) the roles of the control parameters, the ratio, as well as the size of the coupling strength and sigmoid function are analyzed. Finally, for the chemical coupling between two non-identical neurons, the transversal system is given and the effect of two coupling strengths on synchronization is analyzed.
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Affiliation(s)
- Penghe Ge
- Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China
| | - Hongjun Cao
- Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China
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Pena Ramirez J, Arellano-Delgado A, Nijmeijer H. Enhancing master-slave synchronization: The effect of using a dynamic coupling. Phys Rev E 2018; 98:012208. [PMID: 30110856 DOI: 10.1103/physreve.98.012208] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2017] [Indexed: 06/08/2023]
Abstract
This paper introduces a modified master-slave synchronization scheme for dynamical systems. In contrast to the standard configuration, the slave system does not receive any driving signal from the master, but rather the interaction is through a linear dynamical system. The key feature of the proposed coupling scheme is that it induces synchronization in certain systems that cannot be synchronized when using the classical static interconnection. Likewise, the dynamic coupling achieves synchronization for arbitrarily large coupling strength values in certain systems for which the classical configuration is applicable only within a narrow interval of coupling strength values. The performance of the synchronization scheme is illustrated in pairs of identical chaotic and mechanical oscillators.
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Affiliation(s)
- J Pena Ramirez
- Applied Physics Division, Center for Scientific Research and Higher Education at Ensenada (CONACYT-CICESE). Carretera Ensenada-Tijuana 3918, Zona Playitas, Ensenada, Codigo Postal 22860, Baja California, Mexico
| | - A Arellano-Delgado
- Engineering, Architecture, and Design Faculty, Autonomous University of Baja California (CONACYT-UABC). Carretera Ensenada-Tijuana 3917, Zona Playitas, Ensenada, Codigo Postal 22860, Baja California, Mexico
| | - H Nijmeijer
- Department of Mechanical Engineering, Eindhoven University of Technology. P.O. Box 513, 5600 MB Eindhoven, The Netherlands
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Zhuo Z, Cai SM, Tang M, Lai YC. Accurate detection of hierarchical communities in complex networks based on nonlinear dynamical evolution. CHAOS (WOODBURY, N.Y.) 2018; 28:043119. [PMID: 31906645 DOI: 10.1063/1.5025646] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
One of the most challenging problems in network science is to accurately detect communities at distinct hierarchical scales. Most existing methods are based on structural analysis and manipulation, which are NP-hard. We articulate an alternative, dynamical evolution-based approach to the problem. The basic principle is to computationally implement a nonlinear dynamical process on all nodes in the network with a general coupling scheme, creating a networked dynamical system. Under a proper system setting and with an adjustable control parameter, the community structure of the network would "come out" or emerge naturally from the dynamical evolution of the system. As the control parameter is systematically varied, the community hierarchies at different scales can be revealed. As a concrete example of this general principle, we exploit clustered synchronization as a dynamical mechanism through which the hierarchical community structure can be uncovered. In particular, for quite arbitrary choices of the nonlinear nodal dynamics and coupling scheme, decreasing the coupling parameter from the global synchronization regime, in which the dynamical states of all nodes are perfectly synchronized, can lead to a weaker type of synchronization organized as clusters. We demonstrate the existence of optimal choices of the coupling parameter for which the synchronization clusters encode accurate information about the hierarchical community structure of the network. We test and validate our method using a standard class of benchmark modular networks with two distinct hierarchies of communities and a number of empirical networks arising from the real world. Our method is computationally extremely efficient, eliminating completely the NP-hard difficulty associated with previous methods. The basic principle of exploiting dynamical evolution to uncover hidden community organizations at different scales represents a "game-change" type of approach to addressing the problem of community detection in complex networks.
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Affiliation(s)
- Zhao Zhuo
- Web Sciences Center, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Shi-Min Cai
- Web Sciences Center, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Ming Tang
- Institute of Fundamental and Frontier Sciences and Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Ying-Cheng Lai
- School of Electrical Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
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