1
|
Sabhahit NG, Khurd AS, Jalan S. Prolonged hysteresis in the Kuramoto model with inertia and higher-order interactions. Phys Rev E 2024; 109:024212. [PMID: 38491693 DOI: 10.1103/physreve.109.024212] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2023] [Accepted: 01/08/2024] [Indexed: 03/18/2024]
Abstract
The inclusion of inertia in the Kuramoto model has long been reported to change the nature of a phase transition, providing a fertile ground to model the dynamical behaviors of interacting units. More recently, higher-order interactions have been realized as essential for the functioning of real-world complex systems ranging from the brain to disease spreading. Yet analytical insights to decipher the role of inertia with higher-order interactions remain challenging. Here, we study the Kuramoto model with inertia on simplicial complexes, merging two research domains. We develop an analytical framework in a mean-field setting using self-consistent equations to describe the steady-state behavior, which reveals a prolonged hysteresis in the synchronization profile. Inertia and triadic interaction strength exhibit isolated influence on system dynamics by predominantly governing, respectively, the forward and backward transition points. This paper sets a paradigm to deepen our understanding of real-world complex systems such as power grids modeled as the Kuramoto model with inertia.
Collapse
Affiliation(s)
- Narayan G Sabhahit
- Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India
| | - Akanksha S Khurd
- Department of Physics, Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India
| | - Sarika Jalan
- Complex Systems Lab, Department of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore-453552, India
| |
Collapse
|
2
|
Zou W, Chen Y, Senthilkumar DV, Kurths J. Oscillation quenching in diffusively coupled dynamical networks with inertial effects. CHAOS (WOODBURY, N.Y.) 2022; 32:041102. [PMID: 35489855 DOI: 10.1063/5.0087839] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2022] [Accepted: 03/22/2022] [Indexed: 06/14/2023]
Abstract
Self-sustained oscillations are ubiquitous and of fundamental importance for a variety of physical and biological systems including neural networks, cardiac dynamics, and circadian rhythms. In this work, oscillation quenching in diffusively coupled dynamical networks including "inertial" effects is analyzed. By adding inertia to diffusively coupled first-order oscillatory systems, we uncover that even small inertia is capable of eradicating the onset of oscillation quenching. We consolidate the generality of inertia in eradicating oscillation quenching by extensively examining diverse quenching scenarios, where macroscopic oscillations are extremely deteriorated and even completely lost in the corresponding models without inertia. The presence of inertia serves as an additional scheme to eradicate the onset of oscillation quenching, which does not need to tailor the coupling functions. Our findings imply that inertia of a system is an enabler against oscillation quenching in coupled dynamical networks, which, in turn, is helpful for understanding the emergence of rhythmic behaviors in complex coupled systems with amplitude degree of freedom.
Collapse
Affiliation(s)
- Wei Zou
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
| | - Yuxuan Chen
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, Kerala, India
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Telegraphenberg, Potsdam D-14415, Germany
| |
Collapse
|
3
|
Prousalis D, Wetzel L. Synchronization in the presence of time delays and inertia: Stability criteria. Phys Rev E 2022; 105:014210. [PMID: 35193231 DOI: 10.1103/physreve.105.014210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2021] [Accepted: 12/27/2021] [Indexed: 06/14/2023]
Abstract
Linear stability of synchronized states in networks of delay-coupled oscillators depends on the type of interaction, the network, and oscillator properties. For inert oscillator response, found ubiquitously from biology to engineering, states with time-dependent frequencies can arise. These generate side bands in the frequency spectrum or lead to chaotic dynamics. The time delay introduces multistability of synchronized states and an exponential term in the characteristic equation. Stability analysis using the resulting transcendental characteristic equation is a difficult task and is usually carried out numerically. We derive criteria and conditions that enable fast and robust analytical linear stability analysis based on the system parameters. These apply to arbitrary network topologies, identical oscillators, and delays.
Collapse
Affiliation(s)
| | - Lucas Wetzel
- Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
| |
Collapse
|
4
|
Gao J, Efstathiou K. Synchronized clusters in globally connected networks of second-order oscillators: Uncovering the role of inertia. CHAOS (WOODBURY, N.Y.) 2021; 31:093137. [PMID: 34598453 DOI: 10.1063/5.0057125] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2021] [Accepted: 09/01/2021] [Indexed: 06/13/2023]
Abstract
We discuss the formation of secondary synchronized clusters, that is, small clusters of synchronized oscillators besides the main cluster, in second-order oscillator networks and the role of inertia in this process. Such secondary synchronized clusters give rise to non-stationary states such as oscillatory and standing wave states. After describing the formation of such clusters through numerical simulations, we use a time-periodic mean field ansatz to obtain a qualitative understanding of the formation of non-stationary states. Finally, the effect of inertia in the formation of secondary synchronized clusters is analyzed through a minimal model. The analysis shows that the effect of the main synchronized cluster on the other oscillators is weakened by inertias, thus leading to secondary synchronized clusters during the transition to synchronization.
Collapse
Affiliation(s)
- Jian Gao
- Bernoulli Institute for Mathematics, Computer Science, and Artificial Intelligence, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
| | - Konstantinos Efstathiou
- Division of Natural and Applied Sciences and Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, No. 8 Duke Avenue, Kunshan 215316, China
| |
Collapse
|
5
|
Wüster S, Bhavna R. Spatial correlations in a finite-range Kuramoto model. Phys Rev E 2020; 101:052210. [PMID: 32575303 DOI: 10.1103/physreve.101.052210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2019] [Accepted: 04/21/2020] [Indexed: 11/07/2022]
Abstract
We study spatial correlations between oscillator phases in the steady state of a Kuramoto model, in which phase oscillators that are randomly distributed in space interact with constant strength but within a limited range. Such a model could be relevant, for example, in the synchronization of gene expression oscillations in cells, where only oscillations of neighboring cells are coupled through cell-cell contacts. We analytically infer spatial phase-phase correlation functions from the known steady-state distribution of oscillators for the case of homogenous frequencies and show that these can contain information about the range and strength of interactions, provided the noise in the system can be estimated. We suggest a method for the latter, and also explore when correlations appear to be ergodic in this model, which would enable an experimental measurement of correlation functions to utilize temporal averages. Simulations show that our techniques also give qualitative results for the model with heterogenous frequencies. We illustrate our results by comparison with experimental data on genetic oscillations in the segmentation clock of zebrafish embryos.
Collapse
Affiliation(s)
- Sebastian Wüster
- Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 023, India
| | - Rajasekaran Bhavna
- Department of Biological Sciences, Tata Institute of Fundamental Research, 400005 Mumbai, India
| |
Collapse
|
6
|
Kirillov SY, Klinshov VV, Nekorkin VI. The role of timescale separation in oscillatory ensembles with competitive coupling. CHAOS (WOODBURY, N.Y.) 2020; 30:051101. [PMID: 32491880 DOI: 10.1063/5.0009074] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2020] [Accepted: 04/10/2020] [Indexed: 06/11/2023]
Abstract
We study a heterogeneous population consisting of two groups of oscillatory elements, one with attractive and one with repulsive coupling. Moreover, we set different internal timescales for the oscillators of the two groups and concentrate on the role of this timescale separation in the collective behavior. Our results demonstrate that it may significantly modify synchronization properties of the system, and the implications are fundamentally different depending on the ratio between the group timescales. For the slower attractive group, synchronization properties are similar to the case of equal timescales. However, when the attractive group is faster, these properties significantly change and bistability appears. The other collective regimes such as frozen states and solitary states are also shown to be crucially influenced by timescale separation.
Collapse
Affiliation(s)
- S Yu Kirillov
- Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod 603950, Russia
| | - V V Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod 603950, Russia
| | - V I Nekorkin
- Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod 603950, Russia
| |
Collapse
|
7
|
Tumash L, Olmi S, Schöll E. Stability and control of power grids with diluted network topology. CHAOS (WOODBURY, N.Y.) 2019; 29:123105. [PMID: 31893638 DOI: 10.1063/1.5111686] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2019] [Accepted: 11/14/2019] [Indexed: 06/10/2023]
Abstract
We consider sparse random networks of Kuramoto phase oscillators with inertia in order to mimic and investigate the dynamics emerging in high-voltage power grids. The corresponding natural frequencies are assumed to be bimodally Gaussian distributed, thus modeling the distribution of both power generators and consumers, which must be in balance. Our main focus is on the theoretical analysis of the linear stability of the frequency-synchronized state, which is necessary for the stable operation of power grids and the control of unstable synchronous states. We demonstrate by numerical simulations that unstable frequency-synchronized states can be stabilized by feedback control. Further, we extend our study to include stochastic temporal power fluctuations and discuss the interplay of topological disorder and Gaussian white noise for various model configurations and finally demonstrate that our control scheme also works well under the influence of noise. Results for synthetic Erdös-Renyi random networks with low average connectivity and with symmetric or asymmetric bimodal frequency distributions are compared with those obtained by considering a real power grid topology, namely, the grid of Italy.
Collapse
Affiliation(s)
- Liudmila Tumash
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Simona Olmi
- INRIA Sophia Antipolis Méditerranée, 2004 Route des Lucioles, 06902 Valbonne, France
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| |
Collapse
|
8
|
Jeon JH, Kim P. Recurrent synchronization of coupled oscillators with spontaneous phase reformation. CHAOS (WOODBURY, N.Y.) 2018; 28:103113. [PMID: 30384644 DOI: 10.1063/1.5029285] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/12/2018] [Accepted: 09/28/2018] [Indexed: 06/08/2023]
Abstract
Self-organizing and spontaneous breaking are seemingly opposite phenomena and hardly captured in a single model. We develop a second order Kuramoto model with phase-induced damping which shows phase locking together with spontaneous synchrony breaking and reformation. In a relatively large regime where the interacting force and the damping ratio are of the same order, the dynamics of the oscillators alternates in an irregular cycle of synchronization, formation-breaking, and reorganization. While the oscillators keep coming back to phase-locked states, their phase distribution repeatedly reforms. Also, the interevent time between bursty deviation from the synchronization states follows a power-law distribution, which implies that the synchronized states are maintained near a tipping point.
Collapse
Affiliation(s)
- Jong-Ha Jeon
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan Metropolitan City 44919, Republic of Korea
| | - Pilwon Kim
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan Metropolitan City 44919, Republic of Korea
| |
Collapse
|
9
|
Tumash L, Olmi S, Schöll E. Effect of disorder and noise in shaping the dynamics of power grids. ACTA ACUST UNITED AC 2018. [DOI: 10.1209/0295-5075/123/20001] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
|
10
|
Jaros P, Brezetsky S, Levchenko R, Dudkowski D, Kapitaniak T, Maistrenko Y. Solitary states for coupled oscillators with inertia. CHAOS (WOODBURY, N.Y.) 2018; 28:011103. [PMID: 29390619 DOI: 10.1063/1.5019792] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Networks of identical oscillators with inertia can display remarkable spatiotemporal patterns in which one or a few oscillators split off from the main synchronized cluster and oscillate with different averaged frequency. Such "solitary states" are impossible for the classical Kuramoto model with sinusoidal coupling. However, if inertia is introduced, these states represent a solid part of the system dynamics, where each solitary state is characterized by the number of isolated oscillators and their disposition in space. We present system parameter regions for the existence of solitary states in the case of local, non-local, and global network couplings and show that they preserve in both thermodynamic and conservative limits. We give evidence that solitary states arise in a homoclinic bifurcation of a saddle-type synchronized state and die eventually in a crisis bifurcation after essential variation of the parameters.
Collapse
Affiliation(s)
- Patrycja Jaros
- Faculty of Mechanical Engineering, Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Serhiy Brezetsky
- Faculty of Mechanical Engineering, Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Roman Levchenko
- Faculty of Radiophysics, Electronics and Computer Systems, Taras Shevchenko National University of Kyiv, Volodymyrska St. 60, 01030 Kyiv, Ukraine
| | - Dawid Dudkowski
- Faculty of Mechanical Engineering, Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Tomasz Kapitaniak
- Faculty of Mechanical Engineering, Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Yuri Maistrenko
- Faculty of Mechanical Engineering, Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| |
Collapse
|
11
|
Multistable states in a system of coupled phase oscillators with inertia. Sci Rep 2017; 7:42178. [PMID: 28176829 PMCID: PMC5296896 DOI: 10.1038/srep42178] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2016] [Accepted: 01/05/2017] [Indexed: 12/05/2022] Open
Abstract
We investigate the generalized Kuramoto model of globally coupled oscillators with inertia, in which oscillators with positive coupling strength are conformists and oscillators with negative coupling strength are contrarians. We consider the correlation between the coupling strengths of oscillators and the distributions of natural frequencies. Two different types of correlations are studied. It is shown that the model supports multistable synchronized states such as different types of travelling wave states, π state and another type of nonstationary state: an oscillating π state. The phase distribution oscillates in a confined region and the phase difference between conformists and contrarians oscillates around π periodically in the oscillating π state. The different types of travelling wave state may be characterized by the speed of travelling wave and the effective frequencies of oscillators. Finally, the bifurcation diagrams of the model in the parameter space are presented.
Collapse
|
12
|
Barré J, Métivier D. Bifurcations and Singularities for Coupled Oscillators with Inertia and Frustration. PHYSICAL REVIEW LETTERS 2016; 117:214102. [PMID: 27911557 DOI: 10.1103/physrevlett.117.214102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2016] [Indexed: 06/06/2023]
Abstract
We prove that any nonzero inertia, however small, is able to change the nature of the synchronization transition in Kuramoto-like models, either from continuous to discontinuous or from discontinuous to continuous. This result is obtained through an unstable manifold expansion in the spirit of Crawford, which features singularities in the vicinity of the bifurcation. Far from being unwanted artifacts, these singularities actually control the qualitative behavior of the system. Our numerical tests fully support this picture.
Collapse
Affiliation(s)
- J Barré
- Université d'Orléans, CNRS, MAPMO, 45067 Orléans Cedex 2, France, Université Côte d'Azur, CNRS, LJAD, 06108 Nice Cedex 02, France, and Institut Universitaire de France, 75005 Paris, France
| | - D Métivier
- Université Côte d'Azur, CNRS, LJAD, 06108 Nice Cedex 02, France
| |
Collapse
|
13
|
Kryukov AK, Petrov VS, Osipov GV, Kurths J. Multistability of synchronous regimes in rotator ensembles. CHAOS (WOODBURY, N.Y.) 2015; 25:123121. [PMID: 26723160 DOI: 10.1063/1.4938181] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We study collective dynamics in rotator ensembles and focus on the multistability of synchronous regimes in a chain of coupled rotators. We provide a detailed analysis of the number of coexisting regimes and estimate in particular, the synchronization boundary for different types of individual frequency distribution. The number of wave-based regimes coexisting for the same parameters and its dependence on the chain length are estimated. We give an analytical estimation for the synchronization frequency of the in-phase regime for a uniform individual frequency distribution.
Collapse
Affiliation(s)
- A K Kryukov
- Department of Control Theory, Nizhny Novgorod State University, 23, Gagarin Avenue, 603950 Nizhny Novgorod, Russia
| | - V S Petrov
- Department of Control Theory, Nizhny Novgorod State University, 23, Gagarin Avenue, 603950 Nizhny Novgorod, Russia
| | - G V Osipov
- Department of Control Theory, Nizhny Novgorod State University, 23, Gagarin Avenue, 603950 Nizhny Novgorod, Russia
| | - J Kurths
- Institute of Applied Physics, Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia
| |
Collapse
|
14
|
Olmi S. Chimera states in coupled Kuramoto oscillators with inertia. CHAOS (WOODBURY, N.Y.) 2015; 25:123125. [PMID: 26723164 DOI: 10.1063/1.4938734] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
The dynamics of two symmetrically coupled populations of rotators is studied for different values of the inertia. The system is characterized by different types of solutions, which all coexist with the fully synchronized state. At small inertia, the system is no more chaotic and one observes mainly quasi-periodic chimeras, while the usual (stationary) chimera state is not anymore observable. At large inertia, one observes two different kind of chaotic solutions with broken symmetry: the intermittent chaotic chimera, characterized by a synchronized population and a population displaying a turbulent behaviour, and a second state where the two populations are both chaotic but whose dynamics adhere to two different macroscopic attractors. The intermittent chaotic chimeras are characterized by a finite life-time, whose duration increases as a power-law with the system size and the inertia value. Moreover, the chaotic population exhibits clear intermittent behavior, displaying a laminar phase where the two populations tend to synchronize, and a turbulent phase where the macroscopic motion of one population is definitely erratic. In the thermodynamic limit, these states survive for infinite time and the laminar regimes tends to disappear, thus giving rise to stationary chaotic solutions with broken symmetry contrary to what observed for chaotic chimeras on a ring geometry.
Collapse
Affiliation(s)
- Simona Olmi
- CNR - Consiglio Nazionale delle Ricerche - Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
| |
Collapse
|
15
|
Olmi S, Martens EA, Thutupalli S, Torcini A. Intermittent chaotic chimeras for coupled rotators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:030901. [PMID: 26465413 DOI: 10.1103/physreve.92.030901] [Citation(s) in RCA: 38] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2015] [Indexed: 06/05/2023]
Abstract
Two symmetrically coupled populations of N oscillators with inertia m display chaotic solutions with broken symmetry similar to experimental observations with mechanical pendulums. In particular, we report evidence of intermittent chaotic chimeras, where one population is synchronized and the other jumps erratically between laminar and turbulent phases. These states have finite lifetimes diverging as a power law with N and m. Lyapunov analyses reveal chaotic properties in quantitative agreement with theoretical predictions for globally coupled dissipative systems.
Collapse
Affiliation(s)
- Simona Olmi
- Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, I-50019 Sesto Fiorentino, Florence, Italy
- INFN Sezione Firenze, Via Sansone 1, I-50019 Sesto Fiorentino, Florence, Italy
| | - Erik A Martens
- Department of Biomedical Sciences, University of Copenhagen, Blegdamsvej 3, 2200 Copenhagen, Denmark
- Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
- Group of Biophysics and Evolutionary Dynamics, Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Lower Saxony, Germany
| | - Shashi Thutupalli
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Alessandro Torcini
- Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, I-50019 Sesto Fiorentino, Florence, Italy
- INFN Sezione Firenze, Via Sansone 1, I-50019 Sesto Fiorentino, Florence, Italy
- Aix-Marseille Université, Inserm, INMED UMR 901 and Institut de Neurosciences des Systèmes UMR 1106, 13000 Marseille, France
- Aix-Marseille Université, Université de Toulon, CNRS, CPT, UMR 7332, 13288 Marseille, France
| |
Collapse
|
16
|
Ji P, Peron TKDM, Rodrigues FA, Kurths J. Analysis of cluster explosive synchronization in complex networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062810. [PMID: 25615151 DOI: 10.1103/physreve.90.062810] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2014] [Indexed: 06/04/2023]
Abstract
Correlations between intrinsic dynamics and local topology have become a new trend in the study of synchronization in complex networks. In this paper, we investigate the influence of topology on the dynamics of networks made up of second-order Kuramoto oscillators. In particular, based on mean-field calculations, we provide a detailed investigation of cluster explosive synchronization (CES) [Phys. Rev. Lett. 110, 218701 (2013)] in scale-free networks as a function of several topological properties. Moreover, we investigate the robustness of discontinuous transitions by including an additional quenched disorder, and we show that the phase coherence decreases with increasing strength of the quenched disorder. These results complement the previous findings regarding CES and also fundamentally deepen the understanding of the interplay between topology and dynamics under the constraint of correlating natural frequencies and local structure.
Collapse
Affiliation(s)
- Peng Ji
- Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany and Department of Physics, Humboldt University, 12489 Berlin, Germany
| | - Thomas K D M Peron
- Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970 São Carlos, São Paulo, Brazil
| | - Francisco A Rodrigues
- Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970 São Carlos, São Paulo, Brazil
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany and Department of Physics, Humboldt University, 12489 Berlin, Germany and Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| |
Collapse
|
17
|
Olmi S, Navas A, Boccaletti S, Torcini A. Hysteretic transitions in the Kuramoto model with inertia. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042905. [PMID: 25375565 DOI: 10.1103/physreve.90.042905] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2014] [Indexed: 06/04/2023]
Abstract
We report finite-size numerical investigations and mean-field analysis of a Kuramoto model with inertia for fully coupled and diluted systems. In particular, we examine, for a gaussian distribution of the frequencies, the transition from incoherence to coherence for increasingly large system size and inertia. For sufficiently large inertia the transition is hysteretic, and within the hysteretic region clusters of locked oscillators of various sizes and different levels of synchronization coexist. A modification of the mean-field theory developed by Tanaka, Lichtenberg, and Oishi [Physica D 100, 279 (1997)] allows us to derive the synchronization curve associated to each of these clusters. We have also investigated numerically the limits of existence of the coherent and of the incoherent solutions. The minimal coupling required to observe the coherent state is largely independent of the system size, and it saturates to a constant value already for moderately large inertia values. The incoherent state is observable up to a critical coupling whose value saturates for large inertia and for finite system sizes, while in the thermodinamic limit this critical value diverges proportionally to the mass. By increasing the inertia the transition becomes more complex, and the synchronization occurs via the emergence of clusters of whirling oscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the order parameter. We have shown that the transition remains hysteretic even for randomly diluted networks up to a level of connectivity corresponding to a few links per oscillator. Finally, an application to the Italian high-voltage power grid is reported, which reveals the emergence of quasiperiodic oscillations in the order parameter due to the simultaneous presence of many competing whirling clusters.
Collapse
Affiliation(s)
- Simona Olmi
- CNR (Consiglio Nazionale delle Ricerche), Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy and INFN Sez. Firenze, via Sansone, 1, I-50019 Sesto Fiorentino, Italy
| | - Adrian Navas
- Centre for Biomedical Technology (UPM), 28922 Pozuelo de Alarcón, Madrid, Spain
| | - Stefano Boccaletti
- CNR (Consiglio Nazionale delle Ricerche), Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy and INFN Sez. Firenze, via Sansone, 1, I-50019 Sesto Fiorentino, Italy and Centre for Biomedical Technology (UPM), 28922 Pozuelo de Alarcón, Madrid, Spain
| | - Alessandro Torcini
- CNR (Consiglio Nazionale delle Ricerche), Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy and INFN Sez. Firenze, via Sansone, 1, I-50019 Sesto Fiorentino, Italy
| |
Collapse
|
18
|
Ji P, Peron TKD, Menck PJ, Rodrigues FA, Kurths J. Cluster explosive synchronization in complex networks. PHYSICAL REVIEW LETTERS 2013; 110:218701. [PMID: 23745940 DOI: 10.1103/physrevlett.110.218701] [Citation(s) in RCA: 75] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2013] [Revised: 04/29/2013] [Indexed: 06/02/2023]
Abstract
The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter we demonstrate that the nodes in a second-order Kuramoto model perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call cluster explosive synchronization. We provide a rigorous analytical treatment using a mean-field analysis in uncorrelated networks. Our findings are in good agreement with numerical simulations and fundamentally deepen the understanding of microscopic mechanisms toward synchronization.
Collapse
Affiliation(s)
- Peng Ji
- Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany.
| | | | | | | | | |
Collapse
|
19
|
Chandrasekar VK, Sheeba JH, Lakshmanan M. Mass synchronization: occurrence and its control with possible applications to brain dynamics. CHAOS (WOODBURY, N.Y.) 2010; 20:045106. [PMID: 21198118 DOI: 10.1063/1.3527993] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Occurrence of strong or mass synchronization of a large number of neuronal populations in the brain characterizes its pathological states. In order to establish an understanding of the mechanism underlying such pathological synchronization, we present a model of coupled populations of phase oscillators representing the interacting neuronal populations. Through numerical analysis, we discuss the occurrence of mass synchronization in the model, where a source population which gets strongly synchronized drives the target populations onto mass synchronization. We hypothesize and identify a possible cause for the occurrence of such a synchronization, which is so far unknown: Pathological synchronization is caused not just because of the increase in the strength of coupling between the populations but also because of the strength of the strong synchronization of the drive population. We propose a demand controlled method to control this pathological synchronization by providing a delayed feedback where the strength and frequency of the synchronization determine the strength and the time delay of the feedback. We provide an analytical explanation for the occurrence of pathological synchronization and its control in the thermodynamic limit.
Collapse
Affiliation(s)
- V K Chandrasekar
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, Tamilnadu 620 024, India.
| | | | | |
Collapse
|
20
|
Sheeba JH, Chandrasekar VK, Lakshmanan M. Chimera and globally clustered chimera: impact of time delay. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:046203. [PMID: 20481803 DOI: 10.1103/physreve.81.046203] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2009] [Revised: 01/11/2010] [Indexed: 05/29/2023]
Abstract
Following a short report of our preliminary results [Sheeba, Phys. Rev. E 79, 055203(R) (2009)], we present a more detailed study of the effects of coupling delay in diffusively coupled phase oscillator populations. We find that coupling delay induces chimera and globally clustered chimera (GCC) states in delay coupled populations. We show the existence of multiclustered states that act as link between the chimera and the GCC states. A stable GCC state goes through a variety of GCC states, namely, periodic, aperiodic, long- and short-period breathers and becomes unstable GCC leading to global synchronization in the system, on increasing time delay. We provide numerical evidence and theoretical explanations for the above results and discuss possible applications of the observed phenomena.
Collapse
Affiliation(s)
- Jane H Sheeba
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamilnadu, India
| | | | | |
Collapse
|
21
|
Kogan O, Rogers JL, Cross MC, Refael G. Renormalization group approach to oscillator synchronization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:036206. [PMID: 19905199 DOI: 10.1103/physreve.80.036206] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2008] [Indexed: 05/28/2023]
Abstract
We develop a renormalization group method to investigate synchronization clusters in a one-dimensional chain of nearest-neighbor coupled phase oscillators. The method is best suited for chains with strong disorder in the intrinsic frequencies and coupling strengths. The results are compared with numerical simulations of the chain dynamics and good agreement in several characteristics is found. We apply the renormalization group and simulations to Lorentzian distributions of intrinsic frequencies and couplings and investigate the statistics of the resultant cluster sizes and frequencies, as well as the dependence of the characteristic cluster length upon parameters of these Lorentzian distributions.
Collapse
Affiliation(s)
- Oleg Kogan
- Department of Materials Science, California Institute of Technology, 1200 E California Boulevard, Pasadena, California 91125, USA
| | | | | | | |
Collapse
|
22
|
Sheeba JH, Chandrasekar VK, Lakshmanan M. Globally clustered chimera states in delay-coupled populations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:055203. [PMID: 19518512 DOI: 10.1103/physreve.79.055203] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2009] [Revised: 04/02/2009] [Indexed: 05/27/2023]
Abstract
We have identified the existence of globally clustered chimera states in delay-coupled oscillator populations and find that these states can breathe periodically and aperiodically and become unstable depending upon the value of coupling delay. We also find that the coupling delay induces frequency suppression in the desynchronized group. We provide numerical evidence and theoretical explanations for the above results and discuss possible applications of the observed phenomena.
Collapse
Affiliation(s)
- Jane H Sheeba
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, 620024 Tamilnadu, India
| | | | | |
Collapse
|
23
|
Sheeba JH, Chandrasekar VK, Stefanovska A, McClintock PVE. Asymmetry-induced effects in coupled phase-oscillator ensembles: Routes to synchronization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:046210. [PMID: 19518315 DOI: 10.1103/physreve.79.046210] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2008] [Indexed: 05/18/2023]
Abstract
A system of two coupled ensembles of phase oscillators can follow different routes to interensemble synchronization. Following a short report of our preliminary results [Phys. Rev. E 78, 025201(R) (2008)], we present a more detailed study of the effects of coupling, noise, and phase asymmetries in coupled phase-oscillator ensembles. We identify five distinct synchronization regions and routes to synchronization that are characteristic of the coupling asymmetry. We show that noise asymmetry induces effects similar to that of coupling asymmetry when the latter is absent. We also find that phase asymmetry controls the probability of occurrence of particular routes to synchronization. Our results suggest that asymmetry plays a crucial role in controlling synchronization within and between oscillator ensembles, and hence that its consideration is vital for modeling real life problems.
Collapse
Affiliation(s)
- Jane H Sheeba
- Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
| | | | | | | |
Collapse
|
24
|
Sheeba JH, Chandrasekar VK, Stefanovska A, McClintock PVE. Routes to synchrony between asymmetrically interacting oscillator ensembles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:025201. [PMID: 18850880 DOI: 10.1103/physreve.78.025201] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2007] [Revised: 06/20/2008] [Indexed: 05/18/2023]
Abstract
We report that asymmetrically interacting ensembles of oscillators follow novel routes to synchrony. These routes seem to be a characteristic feature of coupling asymmetry. We show that they are unaffected by white noise except that the entrainment frequencies are shifted. The probability of occurrence of the routes is determined by phase asymmetry. The identification of these phenomena offers new insight into synchrony between oscillator ensembles and suggest new ways in which it may be controlled.
Collapse
Affiliation(s)
- Jane H Sheeba
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
| | | | | | | |
Collapse
|
25
|
Guan S, Wang X, Li K, Wang BH, Lai CH. Synchronizability of network ensembles with prescribed statistical properties. CHAOS (WOODBURY, N.Y.) 2008; 18:013120. [PMID: 18377071 DOI: 10.1063/1.2841198] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
It has been shown that synchronizability of a network is determined by the local structure rather than the global properties. With the same global properties, networks may have very different synchronizability. In this paper, we numerically studied, through the spectral properties, the synchronizability of ensembles of networks with prescribed statistical properties. Given a degree sequence, it is found that the eigenvalues and eigenratios characterizing network synchronizability have well-defined distributions, and statistically, the networks with extremely poor synchronizability are rare. Moreover, we compared the synchronizability of three network ensembles that have the same nodes and average degree. Our work reveals that the synchronizability of a network can be significantly affected by the local pattern of connections, and the homogeneity of degree can greatly enhance network synchronizability for networks of a random nature.
Collapse
Affiliation(s)
- Shuguang Guan
- Temasek Laboratories, National University of Singapore, 117508 Singapore
| | | | | | | | | |
Collapse
|
26
|
Filatrella G, Pedersen NF, Wiesenfeld K. Generalized coupling in the Kuramoto model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:017201. [PMID: 17358294 DOI: 10.1103/physreve.75.017201] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2006] [Revised: 09/22/2006] [Indexed: 05/14/2023]
Abstract
We propose a modification of the Kuramoto model to account for the effective change in the coupling constant among the oscillators, as suggested by some experiments on Josephson junction, laser arrays, and mechanical systems, where the active elements are turned on one by one. The resulting model is analytically tractable and predicts that both first and second order phase transitions are possible, depending upon the value of a new parameter that tunes the coupling among the oscillators. Numerical simulations of the model are in accordance with the analytical estimates, and in qualitative agreement with the behavior of Josephson junctions coupled via a cavity.
Collapse
Affiliation(s)
- G Filatrella
- CNR-INFM SuperMat Salerno and Department of Biological and Environmental Sciences, University of Sannio, Via Port'Arsa 11, I-82100, Benevento, Italy
| | | | | |
Collapse
|
27
|
Chacón R. Controlling chaos with localized heterogeneous forces in oscillator chains. Phys Rev E 2006; 74:046202. [PMID: 17155148 DOI: 10.1103/physreve.74.046202] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2006] [Indexed: 11/07/2022]
Abstract
The effects of decreasing the impulse transmitted by localized periodic pulses on the chaotic behavior of homogeneous chains of coupled nonlinear oscillators are studied. It is assumed that when the oscillators are driven synchronously, i.e., all driving pulses transmit the same impulse, the chains display chaotic dynamics. It is shown that decreasing the impulse transmitted by the pulses of the two free end oscillators results in regularization with the whole array exhibiting frequency synchronization, irrespective of the chain size. A maximum level of amplitude desynchrony as the pulses of the two end oscillators narrow is typically found, which is explained as the result of two competing universal mechanisms: desynchronization induced by localized heterogeneous pulses and oscillation death of the complete chain induced by drastic decreasing of the impulse transmitted by such localized pulses. These findings demonstrate that decreasing the impulse transmitted by localized external forces can suppress chaos and lead to frequency-locked states in networks of dissipative systems.
Collapse
Affiliation(s)
- Ricardo Chacón
- Departamento de Física Aplicada, Escuela de Ingenierías Industriales, Universidad de Extremadura, Apartado Postal 382, E-06071 Badajoz, Spain
| |
Collapse
|
28
|
Paoletti MS, Nugent CR, Solomon TH. Synchronization of oscillating reactions in an extended fluid system. PHYSICAL REVIEW LETTERS 2006; 96:124101. [PMID: 16605908 DOI: 10.1103/physrevlett.96.124101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2005] [Indexed: 05/08/2023]
Abstract
We present experiments on the synchronization of a dynamical, chemical process in an extended, flowing, fluid system. The oscillatory Belousov-Zhabotinsky chemical reaction is the process studied, and the flow is an annular chain of counterrotating vortices. Azimuthal motion of the vortices is controlled externally, enabling us to vary the type of transport. We find that oscillations of the Belousov-Zhabotinsky reaction synchronize throughout the extended fluid system only if transport in the flow is superdiffusive, with tracers in the flow undergoing rapid, distant jumps called Lévy flights.
Collapse
Affiliation(s)
- M S Paoletti
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | | | | |
Collapse
|
29
|
Machida M. Effects of edge boundaries on Josephson vortices in finite-size layered high-Tc superconductors. PHYSICAL REVIEW LETTERS 2006; 96:097002. [PMID: 16606298 DOI: 10.1103/physrevlett.96.097002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2005] [Indexed: 05/08/2023]
Abstract
In order to systematically explain the in-plane size effects for the periodical dependence of the Josephson-vortex-flow resistance on the magnetic field, we numerically explore static lattice structures of Josephson vortices in layered high-Tc superconductors with finite in-plane sizes from sub-microm to more than 10 microm by simulating slow quenching processes from high temperature under the magnetic field. The numerical results reveal that in sub-microm size the rectangular lattice is a widely spread major structure in H-T diagram and the triangular lattice is a minor one which emerges only around the specific magnetic field supplying n(phi)0 per one junction area. These results suggest that sub-microm size layered high-Tc superconductors are promising for future device applications.
Collapse
Affiliation(s)
- Masahiko Machida
- CCSE, Japan Atomic Energy Research Institue, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo 110-0015 Japan
| |
Collapse
|
30
|
Wiley DA, Strogatz SH, Girvan M. The size of the sync basin. CHAOS (WOODBURY, N.Y.) 2006; 16:015103. [PMID: 16599769 DOI: 10.1063/1.2165594] [Citation(s) in RCA: 102] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
We suggest a new line of research that we hope will appeal to the nonlinear dynamics community, especially the readers of this Focus Issue. Consider a network of identical oscillators. Suppose the synchronous state is locally stable but not globally stable; it competes with other attractors for the available phase space. How likely is the system to synchronize, starting from a random initial condition? And how does the probability of synchronization depend on the way the network is connected? On the one hand, such questions are inherently difficult because they require calculation of a global geometric quantity, the size of the "sync basin" (or, more formally, the measure of the basin of attraction for the synchronous state). On the other hand, these questions are wide open, important in many real-world settings, and approachable by numerical experiments on various combinations of dynamical systems and network topologies. To give a case study in this direction, we report results on the sync basin for a ring of n >> 1 identical phase oscillators with sinusoidal coupling. Each oscillator interacts equally with its k nearest neighbors on either side. For k/n greater than a critical value (approximately 0.34, obtained analytically), we show that the sync basin is the whole phase space, except for a set of measure zero. As k/n passes below this critical value, coexisting attractors are born in a well-defined sequence. These take the form of uniformly twisted waves, each characterized by an integer winding number q, the number of complete phase twists in one circuit around the ring. The maximum stable twist is proportional to n/k; the constant of proportionality is also obtained analytically. For large values of n/k, corresponding to large rings or short-range coupling, many different twisted states compete for their share of phase space. Our simulations reveal that their basin sizes obey a tantalizingly simple statistical law: the probability that the final state has q twists follows a Gaussian distribution with respect to q. Furthermore, as n/k increases, the standard deviation of this distribution grows linearly with square root of n/k. We have been unable to explain either of these last two results by anything beyond a hand-waving argument.
Collapse
Affiliation(s)
- Daniel A Wiley
- Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
| | | | | |
Collapse
|