1
|
Khatun AA, Muthanna YA, Punetha N, Jafri HH. Collective dynamics of coupled Lorenz oscillators near the Hopf boundary: Intermittency and chimera states. Phys Rev E 2024; 109:034208. [PMID: 38632727 DOI: 10.1103/physreve.109.034208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2022] [Accepted: 02/05/2024] [Indexed: 04/19/2024]
Abstract
We study collective dynamics of networks of mutually coupled identical Lorenz oscillators near a subcritical Hopf bifurcation. Such systems exhibit induced multistable behavior with interesting spatiotemporal dynamics including synchronization, desynchronization, and chimera states. For analysis, we first consider a ring topology with nearest-neighbor coupling and find that the system may exhibit intermittent behavior due to the complex basin structures and dynamical frustration, where temporal dynamics of the oscillators in the ensemble switches between different attractors. Consequently, different oscillators may show a dynamics that is intermittently synchronized (or desynchronized), giving rise to intermittent chimera states. The behavior of the intermittent laminar phases is characterized by the characteristic time spent in the synchronization manifold, which decays as a power law. Such intermittent dynamics is quite general and is also observed in an ensemble of a large number of oscillators arranged in variety of network topologies including nonlocal, scale-free, random, and small-world networks.
Collapse
Affiliation(s)
- Anjuman Ara Khatun
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
- Department of Physics, Indian Institute of Technology Bombay, Mumbai 400 076, India
| | - 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
| |
Collapse
|
2
|
Crnkić A, Jaćimović V. Chimeras and traveling waves in ensembles of Kuramoto oscillators off the Poisson manifold. CHAOS (WOODBURY, N.Y.) 2024; 34:023130. [PMID: 38386905 DOI: 10.1063/5.0184433] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2023] [Accepted: 02/02/2024] [Indexed: 02/24/2024]
Abstract
We examine how perturbations off the Poisson manifold affect chimeras and traveling waves (TWs) in Kuramoto models with two sub-populations. Our numerical study is based on simulations on invariant manifolds, which contain von Mises probability distributions. Our study demonstrates that chimeras and TWs off the Poisson manifold always "breathe", and the effect of breathing is more pronounced further from the Poisson manifold. On the other side, TWs arising in similar models on the sphere always breathe moderately, no matter if the dynamics take place near the Poisson manifold or far away from it.
Collapse
Affiliation(s)
- Aladin Crnkić
- Faculty of Technical Engineering, University of Bihać, I. Ljubijankića, bb., 77000 Bihać, Bosnia and Herzegovina
| | - Vladimir Jaćimović
- Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put, bb., 81000 Podgorica, Montenegro
| |
Collapse
|
3
|
Mishra A, Saha S, Dana SK. Chimeras in globally coupled oscillators: A review. CHAOS (WOODBURY, N.Y.) 2023; 33:092101. [PMID: 37703474 DOI: 10.1063/5.0143872] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 08/21/2023] [Indexed: 09/15/2023]
Abstract
The surprising phenomenon of chimera in an ensemble of identical oscillators is no more strange behavior of network dynamics and reality. By this time, this symmetry breaking self-organized collective dynamics has been established in many networks, a ring of non-locally coupled oscillators, globally coupled networks, a three-dimensional network, and multi-layer networks. A variety of coupling and dynamical models in addition to the phase oscillators has been used for a successful observation of chimera patterns. Experimental verification has also been done using metronomes, pendula, chemical, and opto-electronic systems. The phenomenon has also been shown to appear in small networks, and hence, it is not size-dependent. We present here a brief review of the origin of chimera patterns restricting our discussions to networks of globally coupled identical oscillators only. The history of chimeras in globally coupled oscillators is older than what has been reported in nonlocally coupled phase oscillators much later. We elaborate the story of the origin of chimeras in globally coupled oscillators in a chronological order, within our limitations, and with brief descriptions of the significant contributions, including our personal experiences. We first introduce chimeras in non-locally coupled and other network configurations, in general, and then discuss about globally coupled networks in more detail.
Collapse
Affiliation(s)
- Arindam Mishra
- Department of Physics, National University of Singapore, Singapore 117551
| | - Suman Saha
- Cognitive Brain Dynamics Laboratory, National Brain Research Centre, Gurugram 122051, India
| | - Syamal K Dana
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
- Division of Dynamics, Lodz University of Technology, 90-924 Lodz, Poland
| |
Collapse
|
4
|
Lee S, Krischer K. Heteroclinic switching between chimeras in a ring of six oscillator populations. CHAOS (WOODBURY, N.Y.) 2023; 33:2894497. [PMID: 37276574 DOI: 10.1063/5.0147228] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2023] [Accepted: 05/15/2023] [Indexed: 06/07/2023]
Abstract
In a network of coupled oscillators, a symmetry-broken dynamical state characterized by the coexistence of coherent and incoherent parts can spontaneously form. It is known as a chimera state. We study chimera states in a network consisting of six populations of identical Kuramoto-Sakaguchi phase oscillators. The populations are arranged in a ring, and oscillators belonging to one population are uniformly coupled to all oscillators within the same population and to those in the two neighboring populations. This topology supports the existence of different configurations of coherent and incoherent populations along the ring, but all of them are linearly unstable in most of the parameter space. Yet, chimera dynamics is observed from random initial conditions in a wide parameter range, characterized by one incoherent and five synchronized populations. These observable states are connected to the formation of a heteroclinic cycle between symmetric variants of saddle chimeras, which gives rise to a switching dynamics. We analyze the dynamical and spectral properties of the chimeras in the thermodynamic limit using the Ott-Antonsen ansatz and in finite-sized systems employing Watanabe-Strogatz reduction. For a heterogeneous frequency distribution, a small heterogeneity renders a heteroclinic switching dynamics asymptotically attracting. However, for a large heterogeneity, the heteroclinic orbit does not survive; instead, it is replaced by a variety of attracting chimera states.
Collapse
Affiliation(s)
- Seungjae Lee
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| |
Collapse
|
5
|
Lee S, Krischer K. Chaotic chimera attractors in a triangular network of identical oscillators. Phys Rev E 2023; 107:054205. [PMID: 37328989 DOI: 10.1103/physreve.107.054205] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2022] [Accepted: 04/17/2023] [Indexed: 06/18/2023]
Abstract
A prominent type of collective dynamics in networks of coupled oscillators is the coexistence of coherently and incoherently oscillating domains known as chimera states. Chimera states exhibit various macroscopic dynamics with different motions of the Kuramoto order parameter. Stationary, periodic and quasiperiodic chimeras are known to occur in two-population networks of identical phase oscillators. In a three-population network of identical Kuramoto-Sakaguchi phase oscillators, stationary and periodic symmetric chimeras were previously studied on a reduced manifold in which two populations behaved identically [Phys. Rev. E 82, 016216 (2010)1539-375510.1103/PhysRevE.82.016216]. In this paper, we study the full phase space dynamics of such three-population networks. We demonstrate the existence of macroscopic chaotic chimera attractors that exhibit aperiodic antiphase dynamics of the order parameters. We observe these chaotic chimera states in both finite-sized systems and the thermodynamic limit outside the Ott-Antonsen manifold. The chaotic chimera states coexist with a stable chimera solution on the Ott-Antonsen manifold that displays periodic antiphase oscillation of the two incoherent populations and with a symmetric stationary chimera solution, resulting in tristability of chimera states. Of these three coexisting chimera states, only the symmetric stationary chimera solution exists in the symmetry-reduced manifold.
Collapse
Affiliation(s)
- Seungjae Lee
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| |
Collapse
|
6
|
Jüttner B, Martens EA. Complex dynamics in adaptive phase oscillator networks. CHAOS (WOODBURY, N.Y.) 2023; 33:2888087. [PMID: 37133924 DOI: 10.1063/5.0133190] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2022] [Accepted: 03/28/2023] [Indexed: 05/04/2023]
Abstract
Networks of coupled dynamical units give rise to collective dynamics such as the synchronization of oscillators or neurons in the brain. The ability of the network to adapt coupling strengths between units in accordance with their activity arises naturally in a variety of contexts, including neural plasticity in the brain, and adds an additional layer of complexity: the dynamics on the nodes influence the dynamics of the network and vice versa. We study a minimal model of Kuramoto phase oscillators including a general adaptive learning rule with three parameters (strength of adaptivity, adaptivity offset, adaptivity shift), mimicking learning paradigms based on spike-time-dependent plasticity. Importantly, the strength of adaptivity allows to tune the system away from the limit of the classical Kuramoto model, corresponding to stationary coupling strengths and no adaptation and, thus, to systematically study the impact of adaptivity on the collective dynamics. We carry out a detailed bifurcation analysis for the minimal model consisting of N=2 oscillators. The non-adaptive Kuramoto model exhibits very simple dynamic behavior, drift, or frequency-locking; but once the strength of adaptivity exceeds a critical threshold non-trivial bifurcation structures unravel: A symmetric adaptation rule results in multi-stability and bifurcation scenarios, and an asymmetric adaptation rule generates even more intriguing and rich dynamics, including a period-doubling cascade to chaos as well as oscillations displaying features of both librations and rotations simultaneously. Generally, adaptation improves the synchronizability of the oscillators. Finally, we also numerically investigate a larger system consisting of N=50 oscillators and compare the resulting dynamics with the case of N=2 oscillators.
Collapse
Affiliation(s)
- Benjamin Jüttner
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Erik A Martens
- Centre for Mathematical Sciences, Lund University, Sölvegatan 18B, 221 00 Lund, Sweden
- Center for Translational Neurosciences, University of Copenhagen, Blegdamsvej 3, 2200 Copenhagen, Denmark
| |
Collapse
|
7
|
Ragavan A, Manoranjani M, Senthilkumar DV, Chandrasekar VK. Multistable chimera states in a smallest population of three coupled oscillators. Phys Rev E 2023; 107:044209. [PMID: 37198793 DOI: 10.1103/physreve.107.044209] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Accepted: 03/28/2023] [Indexed: 05/19/2023]
Abstract
We uncover the emergence of distinct sets of multistable chimera states in addition to chimera death and synchronized states in a smallest population of three globally coupled oscillators with mean-field diffusive coupling. Sequence of torus bifurcations result in the manifestation of distinct periodic orbits as a function of the coupling strength, which in turn result in the genesis of distinct chimera states constituted by two synchronized oscillators coexisting with an asynchronous oscillator. Two subsequent Hopf bifurcations result in homogeneous and inhomogeneous steady states resulting in desynchronized steady states and chimera death state among the coupled oscillators. The periodic orbits and the steady states lose their stability via a sequence of saddle-loop and saddle-node bifurcations finally resulting in a stable synchronized state. We have generalized these results to N coupled oscillators and also deduced the variational equations corresponding to the perturbation transverse to the synchronization manifold and corroborated the synchronized state in the two-parameter phase diagrams using its largest eigenvalue. Chimera states in three coupled oscillators emerge as a solitary state in N coupled oscillator ensemble.
Collapse
Affiliation(s)
- A Ragavan
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - M Manoranjani
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - D V Senthilkumar
- School of Physics,Indian Institute of Science Education and Research, Thiruvananthapuram-695551, Kerala, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| |
Collapse
|
8
|
Laing CR. Chimeras on a ring of oscillator populations. CHAOS (WOODBURY, N.Y.) 2023; 33:013121. [PMID: 36725662 DOI: 10.1063/5.0127306] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2022] [Accepted: 12/16/2022] [Indexed: 06/18/2023]
Abstract
Chimeras occur in networks of coupled oscillators and are characterized by coexisting groups of synchronous oscillators and asynchronous oscillators. We consider a network formed from N equal-sized populations at equally spaced points around a ring. We use the Ott/Antonsen ansatz to derive coupled ordinary differential equations governing the level of synchrony within each population and describe chimeras using a self-consistency argument. For N=2 and 3, our results are compared with previously known ones. We obtain new results for the cases of 4,5,…,12 populations and a numerically based conjecture resulting from the behavior of larger numbers of populations. We find macroscopic chaos when more than five populations are considered, but conjecture that this behavior vanishes as the number of populations is increased.
Collapse
Affiliation(s)
- Carlo R Laing
- School of Mathematical and Computational Sciences, Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand
| |
Collapse
|
9
|
Burylko O, Martens EA, Bick C. Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators. CHAOS (WOODBURY, N.Y.) 2022; 32:093109. [PMID: 36182374 DOI: 10.1063/5.0088465] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2022] [Accepted: 08/02/2022] [Indexed: 06/16/2023]
Abstract
Despite their simplicity, networks of coupled phase oscillators can give rise to intriguing collective dynamical phenomena. However, the symmetries of globally and identically coupled identical units do not allow solutions where distinct oscillators are frequency-unlocked-a necessary condition for the emergence of chimeras. Thus, forced symmetry breaking is necessary to observe chimera-type solutions. Here, we consider the bifurcations that arise when full permutational symmetry is broken for the network to consist of coupled populations. We consider the smallest possible network composed of four phase oscillators and elucidate the phase space structure, (partial) integrability for some parameter values, and how the bifurcations away from full symmetry lead to frequency-unlocked weak chimera solutions. Since such solutions wind around a torus they must arise in a global bifurcation scenario. Moreover, periodic weak chimeras undergo a period-doubling cascade leading to chaos. The resulting chaotic dynamics with distinct frequencies do not rely on amplitude variation and arise in the smallest networks that support chaos.
Collapse
Affiliation(s)
- Oleksandr Burylko
- Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str. 3, 01024 Kyiv, Ukraine
| | - Erik A Martens
- Centre for Mathematical Sciences, Lund University, Sölvegatan 18, 221 00 Lund, Sweden
| | - Christian Bick
- Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1111, Amsterdam, The Netherlands
| |
Collapse
|
10
|
Laing CR. Chimeras on annuli. CHAOS (WOODBURY, N.Y.) 2022; 32:083105. [PMID: 36049938 DOI: 10.1063/5.0103669] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2022] [Accepted: 07/08/2022] [Indexed: 06/15/2023]
Abstract
Chimeras occur in networks of coupled oscillators and are characterized by the coexistence of synchronous and asynchronous groups of oscillators in different parts of the network. We consider a network of nonlocally coupled phase oscillators on an annular domain. The Ott/Antonsen ansatz is used to derive a continuum level description of the oscillators' expected dynamics in terms of a complex-valued order parameter. The equations for this order parameter are numerically analyzed in order to investigate solutions with the same symmetry as the domain and chimeras which are analogous to the "multi-headed" chimeras observed on one-dimensional domains. Such solutions are stable only for domains with widths that are neither too large nor too small. We also study rotating waves with different winding numbers, which are similar to spiral wave chimeras seen in two-dimensional domains. We determine ranges of parameters, such as the size of the domain for which such solutions exist and are stable, and the bifurcations by which they lose stability. All of these bifurcations appear subcritical.
Collapse
Affiliation(s)
- Carlo R Laing
- School of Mathematical and Computational Sciences, Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand
| |
Collapse
|
11
|
Laing CR. Chimeras with uniformly distributed heterogeneity: Two coupled populations. Phys Rev E 2022; 105:024306. [PMID: 35291147 DOI: 10.1103/physreve.105.024306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2021] [Accepted: 01/28/2022] [Indexed: 06/14/2023]
Abstract
Chimeras occur in networks of two coupled populations of oscillators when the oscillators in one population synchronize while those in the other are asynchronous. We consider chimeras of this form in networks of planar oscillators for which one parameter associated with the dynamics of an oscillator is randomly chosen from a uniform distribution. A generalization of the previous approach [Laing, Phys. Rev. E 100, 042211 (2019)2470-004510.1103/PhysRevE.100.042211], which dealt with identical oscillators, is used to investigate the existence and stability of chimeras for these heterogeneous networks in the limit of an infinite number of oscillators. In all cases, making the oscillators more heterogeneous destroys the stable chimera in a saddle-node bifurcation. The results help us understand the robustness of chimeras in networks of general oscillators to heterogeneity.
Collapse
Affiliation(s)
- Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Private Bag 102-904 North Shore Mail Centre, Auckland, New Zealand
| |
Collapse
|
12
|
Munyayev VO, Bolotov MI, Smirnov LA, Osipov GV, Belykh IV. Stability of rotatory solitary states in Kuramoto networks with inertia. Phys Rev E 2022; 105:024203. [PMID: 35291064 DOI: 10.1103/physreve.105.024203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2021] [Accepted: 02/03/2022] [Indexed: 06/14/2023]
Abstract
Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and oscillates with a different frequency. Such chimera-type patterns with an incoherent state formed by a single oscillator were observed in various oscillator networks; however, there is still a lack of understanding of how such states can stably appear. Here, we study the stability of solitary states in Kuramoto networks of identical two-dimensional phase oscillators with inertia and a phase-lagged coupling. The presence of inertia can induce rotatory dynamics of the phase difference between the solitary oscillator and the coherent cluster. We derive asymptotic stability conditions for such a solitary state as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (1) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (2) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our stability analysis for the emergence of rotatory chimeras.
Collapse
Affiliation(s)
- Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
- Institute of Applied Physics, Russian Academy of Sciences, Ul'yanova Str. 46, Nizhny Novgorod 603950, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Igor V Belykh
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
- Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302, USA
| |
Collapse
|
13
|
Lee S, Krischer K. Attracting Poisson chimeras in two-population networks. CHAOS (WOODBURY, N.Y.) 2021; 31:113101. [PMID: 34881613 DOI: 10.1063/5.0065710] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2021] [Accepted: 10/11/2021] [Indexed: 06/13/2023]
Abstract
Chimera states, i.e., dynamical states composed of coexisting synchronous and asynchronous oscillations, have been reported to exist in diverse topologies of oscillators in simulations and experiments. Two-population networks with distinct intra- and inter-population coupling have served as simple model systems for chimera states since they possess an invariant synchronized manifold in contrast to networks on a spatial structure. Here, we study dynamical and spectral properties of finite-sized chimeras on two-population networks. First, we elucidate how the Kuramoto order parameter of the finite-sized globally coupled two-population network of phase oscillators is connected to that of the continuum limit. These findings suggest that it is suitable to classify the chimera states according to their order parameter dynamics, and therefore, we define Poisson and non-Poisson chimera states. We then perform a Lyapunov analysis of these two types of chimera states, which yields insight into the full stability properties of the chimera trajectories as well as of collective modes. In particular, our analysis also confirms that Poisson chimeras are neutrally stable. We then introduce two types of "perturbation" that act as small heterogeneities and render Poisson chimeras attracting: A topological variation via the simplest nonlocal intra-population coupling that keeps the network symmetries and the allowance of amplitude variations in the globally coupled two-population network; i.e., we replace the phase oscillators by Stuart-Landau oscillators. The Lyapunov spectral properties of chimera states in the two modified networks are investigated, exploiting an approach based on network symmetry-induced cluster pattern dynamics of the finite-size network.
Collapse
Affiliation(s)
- Seungjae Lee
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| |
Collapse
|
14
|
Barabash NV, Belykh VN, Osipov GV, Belykh IV. Partial synchronization in the second-order Kuramoto model: An auxiliary system method. CHAOS (WOODBURY, N.Y.) 2021; 31:113113. [PMID: 34881584 DOI: 10.1063/5.0066663] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2021] [Accepted: 10/11/2021] [Indexed: 06/13/2023]
Abstract
Partial synchronization emerges in an oscillator network when the network splits into clusters of coherent and incoherent oscillators. Here, we analyze the stability of partial synchronization in the second-order finite-dimensional Kuramoto model of heterogeneous oscillators with inertia. Toward this goal, we develop an auxiliary system method that is based on the analysis of a two-dimensional piecewise-smooth system whose trajectories govern oscillating dynamics of phase differences between oscillators in the coherent cluster. Through a qualitative bifurcation analysis of the auxiliary system, we derive explicit bounds that relate the maximum natural frequency mismatch, inertia, and the network size that can support stable partial synchronization. In particular, we predict threshold-like stability loss of partial synchronization caused by increasing inertia. Our auxiliary system method is potentially applicable to cluster synchronization with multiple coherent clusters and more complex network topology.
Collapse
Affiliation(s)
- Nikita V Barabash
- Department of Mathematics, Volga State University of Water Transport, 5A, Nesterov str., Nizhny Novgorod 603950, Russia
| | - Vladimir N Belykh
- Department of Mathematics, Volga State University of Water Transport, 5A, Nesterov str., Nizhny Novgorod 603950, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23, Gagarin Ave., 603950 Nizhny Novgorod, Russia
| | - Igor V Belykh
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23, Gagarin Ave., 603950 Nizhny Novgorod, Russia
| |
Collapse
|
15
|
Haugland SW, Krischer K. Connecting minimal chimeras and fully asymmetric chaotic attractors through equivariant pitchfork bifurcations. Phys Rev E 2021; 103:L060201. [PMID: 34271668 DOI: 10.1103/physreve.103.l060201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2021] [Accepted: 05/21/2021] [Indexed: 11/06/2022]
Abstract
Highly symmetric networks can exhibit partly symmetry-broken states, including clusters and chimera states, i.e., states of coexisting synchronized and unsynchronized elements. We address the S_{4} permutation symmetry of four globally coupled Stuart-Landau oscillators and uncover an interconnected web of solutions with different symmetries. Among these are chaotic 2-1-1 minimal chimeras that arise from 2-1-1 periodic solutions in a period-doubling cascade, as well as fully asymmetric chaotic states arising similarly from periodic 1-1-1-1 solutions. A backbone of equivariant pitchfork bifurcations mediate between the two cascades, culminating in equivariant pitchforks of chaotic attractors.
Collapse
Affiliation(s)
- Sindre W Haugland
- Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1, D-85748 Garching, Germany
| | - Katharina Krischer
- Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1, D-85748 Garching, Germany
| |
Collapse
|
16
|
Guo S, Yang M, Han W, Yang J. Dynamics in two interacting subpopulations of nonidentical phase oscillators. Phys Rev E 2021; 103:052208. [PMID: 34134272 DOI: 10.1103/physreve.103.052208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/09/2021] [Accepted: 04/26/2021] [Indexed: 11/07/2022]
Abstract
Chimera states refer to the dynamical states in which the inherent symmetry of the system is broken. The system composed of two interacting identical subpopulations of phase oscillators provides a platform to study chimera states. In this system, different types of chimera states have been identified and the transitions between them have been investigated. However, the parameter space is not fully explored in this system. In this work, we study a system comprised of two interacting subpopulations of nonidentical phase oscillators. Through numerical simulations and theoretical analyses, we find three symmetry-reserving states, including incoherent state, in-phase synchronous state, and antiphase synchronous state, and three types of symmetry-breaking states, including in-phase chimera states, antiphase chimera states, and weak chimera states. The stability diagrams of these dynamical states are explored on different parameter planes and transition scenarios amongst these states are investigated. We find that the weak chimera states act as the bridge between in-phase and antiphase chimera states. We also observe the existence of a period-two chimera state, chaotic chimera state, and drifting chimera states.
Collapse
Affiliation(s)
- Shuangjian Guo
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Mingxue Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Wenchen Han
- College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610101, People's Republic of China
| | - Junzhong Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| |
Collapse
|
17
|
Andrzejak RG. Chimeras confined by fractal boundaries in the complex plane. CHAOS (WOODBURY, N.Y.) 2021; 31:053104. [PMID: 34240923 DOI: 10.1063/5.0049631] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2021] [Accepted: 04/16/2021] [Indexed: 06/13/2023]
Abstract
Complex-valued quadratic maps either converge to fixed points, enter into periodic cycles, show aperiodic behavior, or diverge to infinity. Which of these scenarios takes place depends on the map's complex-valued parameter c and the initial conditions. The Mandelbrot set is defined by the set of c values for which the map remains bounded when initiated at the origin of the complex plane. In this study, we analyze the dynamics of a coupled network of two pairs of two quadratic maps in dependence on the parameter c. Across the four maps, c is kept the same whereby the maps are identical. In analogy to the behavior of individual maps, the network iterates either diverge to infinity or remain bounded. The bounded solutions settle into different stable states, including full synchronization and desynchronization of all maps. Furthermore, symmetric partially synchronized states of within-pair synchronization and across-pair synchronization as well as a symmetry broken chimera state are found. The boundaries between bounded and divergent solutions in the domain of c are fractals showing a rich variety of intriguingly esthetic patterns. Moreover, the set of bounded solutions is divided into countless subsets throughout all length scales in the complex plane. Each individual subset contains only one state of synchronization and is enclosed within fractal boundaries by c values leading to divergence.
Collapse
Affiliation(s)
- Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Catalonia, Spain
| |
Collapse
|
18
|
Brezetsky S, Jaros P, Levchenko R, Kapitaniak T, Maistrenko Y. Chimera complexity. Phys Rev E 2021; 103:L050204. [PMID: 34134258 DOI: 10.1103/physreve.103.l050204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2020] [Accepted: 04/23/2021] [Indexed: 06/12/2023]
Abstract
We show an amazing complexity of the chimeras in small networks of coupled phase oscillators with inertia. The network behavior is characterized by heteroclinic switching between multiple saddle chimera states and riddling basins of attractions, causing an extreme sensitivity to initial conditions and parameters. Additional uncertainty is induced by the presumable coexistence of stable phase-locked states or other stable chimeras as the switching trajectories can eventually tend to them. The system dynamics becomes hardly predictable, while its complexity represents a challenge in the network sciences.
Collapse
Affiliation(s)
- Serhiy Brezetsky
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Patrycja Jaros
- Division of Dynamics, Lodz University of Technology, 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
- Forschungszentrum Jülich, 52428 Jülich, Germany
| | - Tomasz Kapitaniak
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Yuri Maistrenko
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
- Forschungszentrum Jülich, 52428 Jülich, Germany
- Institute of Mathematics and Centre for Medical and Biotechnical Research, NAS of Ukraine, Tereshchenkivska St. 3, 01030 Kyiv, Ukraine
| |
Collapse
|
19
|
Zhang Y, Motter AE. Mechanism for Strong Chimeras. PHYSICAL REVIEW LETTERS 2021; 126:094101. [PMID: 33750176 DOI: 10.1103/physrevlett.126.094101] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2020] [Revised: 11/23/2020] [Accepted: 01/27/2021] [Indexed: 06/12/2023]
Abstract
Chimera states have attracted significant attention as symmetry-broken states exhibiting the unexpected coexistence of coherence and incoherence. Despite the valuable insights gained from analyzing specific systems, an understanding of the general physical mechanism underlying the emergence of chimeras is still lacking. Here, we show that many stable chimeras arise because coherence in part of the system is sustained by incoherence in the rest of the system. This mechanism may be regarded as a deterministic analog of noise-induced synchronization and is shown to underlie the emergence of strong chimeras. These are chimera states whose coherent domain is formed by identically synchronized oscillators. Recognizing this mechanism offers a new meaning to the interpretation that chimeras are a natural link between coherence and incoherence.
Collapse
Affiliation(s)
- Yuanzhao Zhang
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, 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
| |
Collapse
|
20
|
Singha J, Gupte N. Chimera states in coupled map lattices: Spatiotemporally intermittent behavior and an equivalent cellular automaton. CHAOS (WOODBURY, N.Y.) 2020; 30:113102. [PMID: 33261350 DOI: 10.1063/5.0016056] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2020] [Accepted: 10/10/2020] [Indexed: 06/12/2023]
Abstract
We construct an equivalent cellular automaton (CA) for a system of globally coupled sine circle maps with two populations and distinct values for intergroup and intragroup coupling. The phase diagram of the system shows that the coupled map lattice can exhibit chimera states with synchronized and spatiotemporally intermittent subgroups after evolution from random initial conditions in some parameter regimes, as well as to other kinds of solutions in other parameter regimes. The CA constructed by us reflects the global nature and the two population structure of the coupled map lattice and is able to reproduce the phase diagram accurately. The CA depends only on the total number of laminar and burst sites and shows a transition from co-existing deterministic and probabilistic behavior in the chimera region to fully probabilistic behavior at the phase boundaries. This identifies the characteristic signature of the transition of a cellular automaton to a chimera state. We also construct an evolution equation for the average number of laminar/burst sites from the CA, analyze its behavior and solutions, and correlate these with the behavior seen for the coupled map lattice. Our CA and methods of analysis can have relevance in wider contexts.
Collapse
Affiliation(s)
- Joydeep Singha
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Neelima Gupte
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| |
Collapse
|
21
|
Suda Y, Okuda K. Emergence of second coherent regions for breathing chimera states. Phys Rev E 2020; 101:062203. [PMID: 32688598 DOI: 10.1103/physreve.101.062203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2019] [Accepted: 04/30/2020] [Indexed: 06/11/2023]
Abstract
Chimera states in one-dimensional nonlocally coupled phase oscillators are mostly assumed to be stationary, but breathing chimeras can occasionally appear, branching from the stationary chimeras via Hopf bifurcation. In this paper, we demonstrate two types of breathing chimeras: The type I breathing chimera looks the same as the stationary chimera at a glance, while the type II consists of multiple coherent regions with different average frequencies. Moreover, it is shown that the type I changes to the type II by increasing the breathing amplitude. Furthermore, we develop a self-consistent analysis of the local order parameter, which can be applied to breathing chimeras, and numerically demonstrate this analysis in the present system.
Collapse
Affiliation(s)
- Yusuke Suda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
- Institute for the Advancement of Higher Education, Hokkaido University, Sapporo 060-0817, Japan
| | - Koji Okuda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
| |
Collapse
|
22
|
Brister BN, Belykh VN, Belykh IV. When three is a crowd: Chaos from clusters of Kuramoto oscillators with inertia. Phys Rev E 2020; 101:062206. [PMID: 32688588 DOI: 10.1103/physreve.101.062206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2020] [Accepted: 06/01/2020] [Indexed: 06/11/2023]
Abstract
Modeling cooperative dynamics using networks of phase oscillators is common practice for a wide spectrum of biological and technological networks, ranging from neuronal populations to power grids. In this paper we study the emergence of stable clusters of synchrony with complex intercluster dynamics in a three-population network of identical Kuramoto oscillators with inertia. The populations have different sizes and can split into clusters where the oscillators synchronize within a cluster, but notably, there is a phase shift between the dynamics of the clusters. We extend our previous results on the bistability of synchronized clusters in a two-population network [I. V. Belykh et al., Chaos 26, 094822 (2016)CHAOEH1054-150010.1063/1.4961435] and demonstrate that the addition of a third population can induce chaotic intercluster dynamics. This effect can be captured by the old adage "two is company, three is a crowd," which suggests that the delicate dynamics of a romantic relationship may be destabilized by the addition of a third party, leading to chaos. Through rigorous analysis and numerics, we demonstrate that the intercluster phase shifts can stably coexist and exhibit different forms of chaotic behavior, including oscillatory, rotatory, and mixed-mode oscillations. We also discuss the implications of our stability results for predicting the emergence of chimeras and solitary states.
Collapse
Affiliation(s)
- Barrett N Brister
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
| | - Vladimir N Belykh
- Department of Mathematics, Volga State University of Water Transport, 5A Nesterov street, Nizhny Novgorod 603950, Russia
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603950, Russia
| | - Igor V Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603950, Russia
| |
Collapse
|
23
|
Bick C, Goodfellow M, Laing CR, Martens EA. Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:9. [PMID: 32462281 PMCID: PMC7253574 DOI: 10.1186/s13408-020-00086-9] [Citation(s) in RCA: 93] [Impact Index Per Article: 23.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 05/07/2020] [Indexed: 05/03/2023]
Abstract
Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.
Collapse
Affiliation(s)
- Christian Bick
- Centre for Systems, Dynamics, and Control, University of Exeter, Exeter, UK.
- Department of Mathematics, University of Exeter, Exeter, UK.
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK.
- Mathematical Institute, University of Oxford, Oxford, UK.
- Institute for Advanced Study, Technische Universität München, Garching, Germany.
| | - Marc Goodfellow
- Department of Mathematics, University of Exeter, Exeter, UK
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK
- Living Systems Institute, University of Exeter, Exeter, UK
- Wellcome Trust Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter, UK
| | - Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark.
- Department of Biomedical Science, University of Copenhagen, Copenhagen N, Denmark.
- Centre for Translational Neuroscience, University of Copenhagen, Copenhagen N, Denmark.
| |
Collapse
|
24
|
Choe CU, Choe MH, Jang H, Kim RS. Symmetry breakings in two populations of oscillators coupled via diffusive environments: Chimera and heterosynchrony. Phys Rev E 2020; 101:042213. [PMID: 32422840 DOI: 10.1103/physreve.101.042213] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2019] [Accepted: 03/16/2020] [Indexed: 11/07/2022]
Abstract
We consider two diffusively coupled populations of identical oscillators, where the oscillators in each population are coupled with a common dynamic environment. Existence and stability of a variety of stationary states are analyzed on the basis of the Ott-Antonsen reduction method, which reveals that the chimera state occurs under the diffusive coupling scheme. Furthermore, we find an exotic symmetry-breaking behavior, the so-called the heterosynchronous state, in which each population exhibits in-phase coherence, while the order parameters of two populations rotate at different phase velocities. The chimera and heterosynchronous states emerge from bistabilities of distinct states for decoupled population and occur as a unique continuation for weak diffusive couplings. The heterosynchronous state is caused by an indirect coupling scheme via dynamic environments and could occur for a finite-size system as well, even for the system that consists of one oscillator per population.
Collapse
Affiliation(s)
- Chol-Ung Choe
- Research Group for Nonlinear Dynamics, Department of Physics, University of Science, Unjong-District, Pyongyang, Democratic People's Republic of Korea
| | - Myong-Hui Choe
- Department of Mathematics, Pyongyang University of Railways, Hyongjesan-District, Pyongyang, Democratic People's Republic of Korea
| | - Hyok Jang
- Research Group for Nonlinear Dynamics, Department of Physics, University of Science, Unjong-District, Pyongyang, Democratic People's Republic of Korea
| | - Ryong-Son Kim
- Research Group for Nonlinear Dynamics, Department of Physics, University of Science, Unjong-District, Pyongyang, Democratic People's Republic of Korea
| |
Collapse
|
25
|
Andrzejak RG, Ruzzene G, Schöll E, Omelchenko I. Two populations of coupled quadratic maps exhibit a plentitude of symmetric and symmetry broken dynamics. CHAOS (WOODBURY, N.Y.) 2020; 30:033125. [PMID: 32237754 DOI: 10.1063/5.0002272] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2020] [Accepted: 02/28/2020] [Indexed: 06/11/2023]
Abstract
We numerically study a network of two identical populations of identical real-valued quadratic maps. Upon variation of the coupling strengths within and across populations, the network exhibits a rich variety of distinct dynamics. The maps in individual populations can be synchronized or desynchronized. Their temporal evolution can be periodic or aperiodic. Furthermore, one can find blends of synchronized with desynchronized states and periodic with aperiodic motions. We show symmetric patterns for which both populations have the same type of dynamics as well as chimera states of a broken symmetry. The network can furthermore show multistability by settling to distinct dynamics for different realizations of random initial conditions or by switching intermittently between distinct dynamics for the same realization. We conclude that our system of two populations of a particularly simple map is the most simple system that can show this highly diverse and complex behavior, which includes but is not limited to chimera states. As an outlook to future studies, we explore the stability of two populations of quadratic maps with a complex-valued control parameter. We show that bounded and diverging dynamics are separated by fractal boundaries in the complex plane of this control parameter.
Collapse
Affiliation(s)
- Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| |
Collapse
|
26
|
Senthilkumar DV, Chandrasekar VK. Local and global chimera states in a four-oscillator system. Phys Rev E 2019; 100:032211. [PMID: 31639927 DOI: 10.1103/physreve.100.032211] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2019] [Indexed: 11/07/2022]
Abstract
Emergence of distinct collective dynamical states in the two smallest populations of nonlinear oscillators with global coupling among them is delineated. In particular, considering four oscillators constituting two populations of two oscillators each, we show the emergence of local chimera, global chimera, and global amplitude chimera states by breaking the permutation symmetry within the populations, in addition to the local synchronized and global synchronized states. Further, the symmetry breaking facilitates the onset of local chimera accompanied by the anomalous synchronization even with an identical nonisochronicity parameter in contrast to the existing literature. Furthermore, the spread of local synchronized and local chimera states is found to increase upon increasing the frequency difference among the populations, while the spread of local chimera, global chimera, and global amplitude chimera states increases for larger values of the nonisochronicity parameter. In addition, we have also deduced the stability condition for the global synchronized state using both the amplitude and its phase model, and find that the global synchronized state attains stability even for smaller values of the coupling strength for the amplitude model when compared to its phase reduced model.
Collapse
Affiliation(s)
- D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695 551, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| |
Collapse
|
27
|
Laing CR. Dynamics and stability of chimera states in two coupled populations of oscillators. Phys Rev E 2019; 100:042211. [PMID: 31770875 DOI: 10.1103/physreve.100.042211] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2019] [Indexed: 06/10/2023]
Abstract
We consider networks formed from two populations of identical oscillators, with uniform strength all-to-all coupling within populations and also between populations with a different strength. Such systems are known to support chimera states in which oscillators within one population are perfectly synchronized while in the other the oscillators are incoherent and have a different mean frequency from those in the synchronous population. Assuming that the oscillators in the incoherent population always lie on a closed smooth curve C, we derive and analyze the dynamics of the shape of C and the probability density on C for four different types of oscillators. We put some previously derived results on a more rigorous footing and analyze two new systems.
Collapse
Affiliation(s)
- Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland 0745, New Zealand
| |
Collapse
|
28
|
Hart JD, Larger L, Murphy TE, Roy R. Delayed dynamical systems: networks, chimeras and reservoir computing. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20180123. [PMID: 31329059 PMCID: PMC6661333 DOI: 10.1098/rsta.2018.0123] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 04/03/2019] [Indexed: 05/23/2023]
Abstract
We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research on optoelectronic feedback loops has revealed their immense potential for realizing complex system dynamics such as chimeras in rings of coupled oscillators and applications to reservoir computing. Delayed dynamical systems have been enriched in recent years through the application of digital signal processing techniques. Very recently, we have showed that one can significantly extend the capabilities and implement networks with arbitrary topologies through the use of field programmable gate arrays. This architecture allows the design of appropriate filters and multiple time delays, and greatly extends the possibilities for exploring synchronization patterns in arbitrary network topologies. This has enabled us to explore complex dynamics on networks with nodes that can be perfectly identical, introduce parameter heterogeneities and multiple time delays, as well as change network topologies to control the formation and evolution of patterns of synchrony. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.
Collapse
Affiliation(s)
- Joseph D. Hart
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA
- Department of Physics, University of Maryland, College Park, MD 20742, USA
| | - Laurent Larger
- FEMTO-ST Institute/Optics Department, CNRS and University Bourgogne Franche-Comté, 15B avenue des Montboucons, 25030 Besançon Cedex, France
| | - Thomas E. Murphy
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA
- Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA
| | - Rajarshi Roy
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA
- Department of Physics, University of Maryland, College Park, MD 20742, USA
- Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
| |
Collapse
|
29
|
Hajnová V, Přibylová L. Bifurcation manifolds in predator-prey models computed by Gröbner basis method. Math Biosci 2019; 312:1-7. [PMID: 30946845 DOI: 10.1016/j.mbs.2019.03.008] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2018] [Revised: 03/30/2019] [Accepted: 03/31/2019] [Indexed: 11/28/2022]
Abstract
Many natural processes studied in population biology, systems biology, biochemistry, chemistry or physics are modeled by dynamical systems with polynomial or rational right-hand sides in state and parameter variables. The problem of finding bifurcation manifolds of such discrete or continuous dynamical systems leads to a problem of finding solutions to a system of non-linear algebraic equations. This approach often fails since it is not possible to express equilibria explicitly. Here we describe an algebraic procedure based on the Gröbner basis computation that finds bifurcation manifolds without computing equilibria. Our method provides formulas for bifurcation manifolds in commonly studied cases in applied research - for the fold, transcritical, cusp, Hopf and Bogdanov-Takens bifurcations. The method returns bifurcation manifolds as implicitly defined functions or parametric functions in full parameter space. The approach can be implemented in any computer algebra system; therefore it can be used in applied research as a supporting autonomous computation even by non-experts in bifurcation theory. This paper demonstrates our new approach on the recently published Rosenzweig-MacArthur predator-prey model generalizations in order to highlight the simplicity of our method compared to the published analysis.
Collapse
Affiliation(s)
- Veronika Hajnová
- Department of Mathematics and Statistics, Section of Applied Mathematics, Faculty of Science, Masaryk University, Kotlářská 2, Brno, 611 37, Czechia.
| | - Lenka Přibylová
- Department of Mathematics and Statistics, Section of Applied Mathematics, Faculty of Science, Masaryk University, Kotlářská 2, Brno, 611 37, Czechia
| |
Collapse
|
30
|
Matheny MH, Emenheiser J, Fon W, Chapman A, Salova A, Rohden M, Li J, Hudoba de Badyn M, Pósfai M, Duenas-Osorio L, Mesbahi M, Crutchfield JP, Cross MC, D’Souza RM, Roukes ML. Exotic states in a simple network of nanoelectromechanical oscillators. Science 2019; 363:363/6431/eaav7932. [DOI: 10.1126/science.aav7932] [Citation(s) in RCA: 82] [Impact Index Per Article: 16.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2018] [Accepted: 01/24/2019] [Indexed: 01/10/2023]
Abstract
Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.
Collapse
|
31
|
Bolotov MI, Munyaev VO, Kryukov AK, Smirnov LA, Osipov GV. Variety of rotation modes in a small chain of coupled pendulums. CHAOS (WOODBURY, N.Y.) 2019; 29:033109. [PMID: 30927845 DOI: 10.1063/1.5079499] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2018] [Accepted: 02/11/2019] [Indexed: 06/09/2023]
Abstract
This article studies the rotational dynamics of three identical coupled pendulums. There exist two parameter areas where the in-phase rotational motion is unstable and out-of-phase rotations are realized. Asymptotic theory is developed that allows us to analytically identify borders of instability areas of in-phase rotation motion. It is shown that out-of-phase rotations are the result of the parametric instability of in-phase motion. Complex out-of-phase rotations are numerically found and their stability and bifurcations are defined. It is demonstrated that the emergence of chaotic dynamics happens due to the period doubling bifurcation cascade. The detailed scenario of symmetry breaking is presented. The development of chaotic dynamics leads to the origin of two chaotic attractors of different types. The first one is characterized by the different phases of all pendulums. In the second case, the phases of the two pendulums are equal, and the phase of the third one is different. This regime can be interpreted as a drum-head mode in star-networks. It may also indicate the occurrence of chimera states in chains with a greater number of nearest-neighbour interacting elements and in analogical systems with global coupling.
Collapse
Affiliation(s)
- Maxim I Bolotov
- Department of Control Theory, Nizhny Novgorod State University, Gagarin Ave. 23, Nizhny Novgorod 603950, Russia
| | - Vyacheslav O Munyaev
- Department of Control Theory, Nizhny Novgorod State University, Gagarin Ave. 23, Nizhny Novgorod 603950, Russia
| | - Alexey K Kryukov
- Department of Control Theory, Nizhny Novgorod State University, Gagarin Ave. 23, Nizhny Novgorod 603950, Russia
| | - Lev A Smirnov
- Department of Control Theory, Nizhny Novgorod State University, Gagarin Ave. 23, Nizhny Novgorod 603950, Russia
| | - Grigory V Osipov
- Department of Control Theory, Nizhny Novgorod State University, Gagarin Ave. 23, Nizhny Novgorod 603950, Russia
| |
Collapse
|
32
|
Kim RS, Choe CU. Chimera state on a spherical surface of nonlocally coupled oscillators with heterogeneous phase lags. CHAOS (WOODBURY, N.Y.) 2019; 29:023101. [PMID: 30823720 DOI: 10.1063/1.5079472] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2018] [Accepted: 01/08/2019] [Indexed: 06/09/2023]
Abstract
We consider a network of coupled oscillators embedded in the surface of a sphere with nonlocal coupling strength and heterogeneous phase lags. A nonlocal coupling scheme with heterogeneous phase lags that allows the system to be solved analytically is suggested and the main effects of heterogeneity in the phase lags on the existence and stability of steady states are analyzed. We explore the stability of solutions along the Ott-Antonsen invariant manifold and present a complete bifurcation diagram for stationary patterns including the coherent, incoherent, and modulated drift states as well as chimera state. The stability analysis shows that a continuum of uniform drift states and the modulated drift state could become stable only due to the heterogeneity of the phase lags and that the chimera state is bifurcated from the modulated drift state. Our theoretical results are verified by using the direct numerical simulations of the model system.
Collapse
Affiliation(s)
- Ryong-Son Kim
- Research Group for Nonlinear Dynamics, Department of Physics, University of Science, Unjong-District, Pyongyang, Democratic People's Republic of Korea
| | - Chol-Ung Choe
- Research Group for Nonlinear Dynamics, Department of Physics, University of Science, Unjong-District, Pyongyang, Democratic People's Republic of Korea
| |
Collapse
|
33
|
Banerjee T, Biswas D, Ghosh D, Schöll E, Zakharova A. Networks of coupled oscillators: From phase to amplitude chimeras. CHAOS (WOODBURY, N.Y.) 2018; 28:113124. [PMID: 30501215 DOI: 10.1063/1.5054181] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2018] [Accepted: 10/31/2018] [Indexed: 06/09/2023]
Abstract
We show that amplitude-mediated phase chimeras and amplitude chimeras can occur in the same network of nonlocally coupled identical oscillators. These are two different partial synchronization patterns, where spatially coherent domains coexist with incoherent domains and coherence/incoherence referring to both amplitude and phase or only the amplitude of the oscillators, respectively. By changing the coupling strength, the two types of chimera patterns can be induced. We find numerically that the amplitude chimeras are not short-living transients but can have a long lifetime. Also, we observe variants of the amplitude chimeras with quasiperiodic temporal oscillations. We provide a qualitative explanation of the observed phenomena in the light of symmetry breaking bifurcation scenarios. We believe that this study will shed light on the connection between two disparate chimera states having different symmetry-breaking properties.
Collapse
Affiliation(s)
- Tanmoy Banerjee
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan, 713 104 West Bengal, India
| | - Debabrata Biswas
- Department of Physics, Rampurhat College, Birbhum, 731 224 West Bengal, India
| | - Debarati Ghosh
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan, 713 104 West Bengal, India
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| |
Collapse
|
34
|
Thoubaan M, Ashwin P. Existence and stability of chimera states in a minimal system of phase oscillators. CHAOS (WOODBURY, N.Y.) 2018; 28:103121. [PMID: 30384640 DOI: 10.1063/1.5044750] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2018] [Accepted: 10/09/2018] [Indexed: 06/08/2023]
Abstract
We examine partial frequency locked weak chimera states in a network of six identical and indistinguishable phase oscillators with neighbour and next-neighbour coupling and two harmonic coupling of the form g ( ϕ ) = - sin ( ϕ - α ) + r sin 2 ϕ . We limit to a specific partial cluster subspace, reduce to a two dimensional system in terms of phase differences, and show that this has an integral of motion for α = π / 2 and r = 0 . By careful analysis of the phase space, we show that there is a continuum of neutrally stable weak chimera states in this case. We approximate the Poincaré return map for these weak chimera solutions and demonstrate several results about the stability and bifurcation of weak chimeras for small β = π / 2 - α and r that agree with numerical path-following of the solutions.
Collapse
Affiliation(s)
- Mary Thoubaan
- Centre for Systems, Dynamics and Control, Department of Mathematics, University of Exeter, Exeter EX4 4QF, United Kingdom
| | - Peter Ashwin
- Centre for Systems, Dynamics and Control, Department of Mathematics, University of Exeter, Exeter EX4 4QF, United Kingdom
| |
Collapse
|
35
|
Majhi S, Ghosh D. Alternating chimeras in networks of ephaptically coupled bursting neurons. CHAOS (WOODBURY, N.Y.) 2018; 28:083113. [PMID: 30180636 DOI: 10.1063/1.5022612] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2018] [Accepted: 07/26/2018] [Indexed: 06/08/2023]
Abstract
The distinctive phenomenon of the chimera state has been explored in neuronal systems under a variety of different network topologies during the last decade. Nevertheless, in all the works, the neurons are presumed to interact with each other directly with the help of synapses only. But, the influence of ephaptic coupling, particularly magnetic flux across the membrane, is mostly unexplored and should essentially be dealt with during the emergence of collective electrical activities and propagation of signals among the neurons in a network. Through this article, we report the development of an emerging dynamical state, namely, the alternating chimera, in a network of identical neuronal systems induced by an external electromagnetic field. Owing to this interaction scenario, the nonlinear neuronal oscillators are coupled indirectly via electromagnetic induction with magnetic flux, through which neurons communicate in spite of the absence of physical connections among them. The evolution of each neuron, here, is described by the three-dimensional Hindmarsh-Rose dynamics. We demonstrate that the presence of such non-locally and globally interacting external environments induces a stationary alternating chimera pattern in the ensemble of neurons, whereas in the local coupling limit, the network exhibits a transient chimera state whenever the local dynamics of the neurons is of the chaotic square-wave bursting type. For periodic square-wave bursting of the neurons, a similar qualitative phenomenon has been witnessed with the exception of the disappearance of cluster states for non-local and global interactions. Besides these observations, we advance our work while providing confirmation of the findings for neuronal ensembles exhibiting plateau bursting dynamics and also put forward the fact that the plateau pattern actually favors the alternating chimera more than others. These results may deliver better interpretations for different aspects of synchronization appearing in a network of neurons through field coupling that also relaxes the prerequisite of synaptic connectivity for realizing the chimera state in neuronal networks.
Collapse
Affiliation(s)
- Soumen Majhi
- 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
| |
Collapse
|
36
|
Bick C, Panaggio MJ, Martens EA. Chaos in Kuramoto oscillator networks. CHAOS (WOODBURY, N.Y.) 2018; 28:071102. [PMID: 30070510 DOI: 10.1063/1.5041444] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2018] [Accepted: 06/20/2018] [Indexed: 05/20/2023]
Abstract
Kuramoto oscillators are widely used to explain collective phenomena in networks of coupled oscillatory units. We show that simple networks of two populations with a generic coupling scheme, where both coupling strengths and phase lags between and within populations are distinct, can exhibit chaotic dynamics as conjectured by Ott and Antonsen [Chaos 18, 037113 (2008)]. These chaotic mean-field dynamics arise universally across network size, from the continuum limit of infinitely many oscillators down to very small networks with just two oscillators per population. Hence, complicated dynamics are expected even in the simplest description of oscillator networks.
Collapse
Affiliation(s)
- Christian Bick
- Department of Mathematics and Centre for Systems Dynamics and Control, University of Exeter, Exeter EX4 4QF, UK
| | - Mark J Panaggio
- Department of Mathematics, Hillsdale College, 33 E College Street, Hillsdale, Michigan 49242, USA
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science,Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| |
Collapse
|
37
|
Kemeth FP, Haugland SW, Krischer K. Symmetries of Chimera States. PHYSICAL REVIEW LETTERS 2018; 120:214101. [PMID: 29883158 DOI: 10.1103/physrevlett.120.214101] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2018] [Revised: 04/18/2018] [Indexed: 05/20/2023]
Abstract
Symmetry broken states arise naturally in oscillatory networks. In this Letter, we investigate chaotic attractors in an ensemble of four mean-coupled Stuart-Landau oscillators with two oscillators being synchronized. We report that these states with partially broken symmetry, so-called chimera states, have different setwise symmetries in the incoherent oscillators, and in particular, some are and some are not invariant under a permutation symmetry on average. This allows for a classification of different chimera states in small networks. We conclude our report with a discussion of related states in spatially extended systems, which seem to inherit the symmetry properties of their counterparts in small networks.
Collapse
Affiliation(s)
- Felix P Kemeth
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, James-Franck-Strasse 1, D-85748 Garching, Germany
- Institute for Advanced Study - Technische Universität München, Lichtenbergstrasse 2a, D-85748 Garching, Germany
| | - Sindre W Haugland
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, James-Franck-Strasse 1, D-85748 Garching, Germany
- Institute for Advanced Study - Technische Universität München, Lichtenbergstrasse 2a, D-85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Nonequilibrium Chemical Physics, Technische Universität München, James-Franck-Strasse 1, D-85748 Garching, Germany
| |
Collapse
|
38
|
Suda Y, Okuda K. Breathing multichimera states in nonlocally coupled phase oscillators. Phys Rev E 2018; 97:042212. [PMID: 29758692 DOI: 10.1103/physreve.97.042212] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2017] [Indexed: 06/08/2023]
Abstract
Chimera states for the one-dimensional array of nonlocally coupled phase oscillators in the continuum limit are assumed to be stationary states in most studies, but a few studies report the existence of breathing chimera states. We focus on multichimera states with two coherent and incoherent regions and numerically demonstrate that breathing multichimera states, whose global order parameter oscillates temporally, can appear. Moreover, we show that the system exhibits a Hopf bifurcation from a stationary multichimera to a breathing one by the linear stability analysis for the stationary multichimera.
Collapse
Affiliation(s)
- Yusuke Suda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
| | - Koji Okuda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
| |
Collapse
|
39
|
Chouzouris T, Omelchenko I, Zakharova A, Hlinka J, Jiruska P, Schöll E. Chimera states in brain networks: Empirical neural vs. modular fractal connectivity. CHAOS (WOODBURY, N.Y.) 2018; 28:045112. [PMID: 31906648 DOI: 10.1063/1.5009812] [Citation(s) in RCA: 46] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Abstract
Complex spatiotemporal patterns, called chimera states, consist of coexisting coherent and incoherent domains and can be observed in networks of coupled oscillators. The interplay of synchrony and asynchrony in complex brain networks is an important aspect in studies of both the brain function and disease. We analyse the collective dynamics of FitzHugh-Nagumo neurons in complex networks motivated by its potential application to epileptology and epilepsy surgery. We compare two topologies: an empirical structural neural connectivity derived from diffusion-weighted magnetic resonance imaging and a mathematically constructed network with modular fractal connectivity. We analyse the properties of chimeras and partially synchronized states and obtain regions of their stability in the parameter planes. Furthermore, we qualitatively simulate the dynamics of epileptic seizures and study the influence of the removal of nodes on the network synchronizability, which can be useful for applications to epileptic surgery.
Collapse
Affiliation(s)
- Teresa Chouzouris
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Jaroslav Hlinka
- Institute of Computer Science, Czech Academy of Sciences, Pod Vodarenskou vezi 2, 18207 Prague, Czech Republic
| | - Premysl Jiruska
- Institute of Physiology, Czech Academy of Sciences, Videnska 1083, 14220 Prague, Czech Republic
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| |
Collapse
|
40
|
Premalatha K, Chandrasekar VK, Senthilvelan M, Lakshmanan M. Stable amplitude chimera states in a network of locally coupled Stuart-Landau oscillators. CHAOS (WOODBURY, N.Y.) 2018; 28:033110. [PMID: 29604660 DOI: 10.1063/1.5006454] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We investigate the occurrence of collective dynamical states such as transient amplitude chimera, stable amplitude chimera, and imperfect breathing chimera states in a locally coupled network of Stuart-Landau oscillators. In an imperfect breathing chimera state, the synchronized group of oscillators exhibits oscillations with large amplitudes, while the desynchronized group of oscillators oscillates with small amplitudes, and this behavior of coexistence of synchronized and desynchronized oscillations fluctuates with time. Then, we analyze the stability of the amplitude chimera states under various circumstances, including variations in system parameters and coupling strength, and perturbations in the initial states of the oscillators. For an increase in the value of the system parameter, namely, the nonisochronicity parameter, the transient chimera state becomes a stable chimera state for a sufficiently large value of coupling strength. In addition, we also analyze the stability of these states by perturbing the initial states of the oscillators. We find that while a small perturbation allows one to perturb a large number of oscillators resulting in a stable amplitude chimera state, a large perturbation allows one to perturb a small number of oscillators to get a stable amplitude chimera state. We also find the stability of the transient and stable amplitude chimera states and traveling wave states for an appropriate number of oscillators using Floquet theory. In addition, we also find the stability of the incoherent oscillation death states.
Collapse
Affiliation(s)
- K Premalatha
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamil Nadu, India
| | - M Senthilvelan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| |
Collapse
|
41
|
Laing CR. The Dynamics of Networks of Identical Theta Neurons. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2018; 8:4. [PMID: 29404814 PMCID: PMC5799134 DOI: 10.1186/s13408-018-0059-7] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2017] [Accepted: 01/26/2018] [Indexed: 05/20/2023]
Abstract
We consider finite and infinite all-to-all coupled networks of identical theta neurons. Two types of synaptic interactions are investigated: instantaneous and delayed (via first-order synaptic processing). Extensive use is made of the Watanabe/Strogatz (WS) ansatz for reducing the dimension of networks of identical sinusoidally-coupled oscillators. As well as the degeneracy associated with the constants of motion of the WS ansatz, we also find continuous families of solutions for instantaneously coupled neurons, resulting from the reversibility of the reduced model and the form of the synaptic input. We also investigate a number of similar related models. We conclude that the dynamics of networks of all-to-all coupled identical neurons can be surprisingly complicated.
Collapse
Affiliation(s)
- Carlo R Laing
- Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand.
| |
Collapse
|
42
|
Ratas I, Pyragas K. Symmetry breaking in two interacting populations of quadratic integrate-and-fire neurons. Phys Rev E 2018; 96:042212. [PMID: 29347512 DOI: 10.1103/physreve.96.042212] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2017] [Indexed: 11/07/2022]
Abstract
We analyze the dynamics of two coupled identical populations of quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The populations are heterogeneous; they include both inherently spiking and excitable neurons. The coupling within and between the populations is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rates and the mean membrane potentials in both populations. The reduced equations are exact in the infinite-size limit. The bifurcation analysis of the equations reveals a rich variety of nonsymmetric patterns, including a splay state, antiphase periodic oscillations, chimera-like states, and chaotic oscillations as well as bistabilities between various states. The validity of the reduced equations is confirmed by direct numerical simulations of the finite-size networks.
Collapse
Affiliation(s)
- Irmantas Ratas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Kestutis Pyragas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| |
Collapse
|
43
|
Proskurkin IS, Vanag VK. Dynamics of a 1D array of inhibitory coupled chemical oscillators in microdroplets with global negative feedback. Phys Chem Chem Phys 2018; 20:16126-16137. [DOI: 10.1039/c8cp02283f] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023]
Abstract
We have investigated the effect of global negative feedback (GNF) on the dynamics of a 1D array of water microdroplets (MDs) filled with the reagents of the photosensitive oscillatory Belousov–Zhabotinsky (BZ) reaction.
Collapse
Affiliation(s)
- Ivan S. Proskurkin
- Centre for Nonlinear Chemistry
- Immanuel Kant Baltic Federal University
- Kaliningrad
- Russia
| | - Vladimir K. Vanag
- Centre for Nonlinear Chemistry
- Immanuel Kant Baltic Federal University
- Kaliningrad
- Russia
| |
Collapse
|
44
|
Omelchenko I, Omel'chenko OE, Zakharova A, Schöll E. Optimal design of tweezer control for chimera states. Phys Rev E 2018; 97:012216. [PMID: 29448470 DOI: 10.1103/physreve.97.012216] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2017] [Indexed: 06/08/2023]
Abstract
Chimera states are complex spatio-temporal patterns which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyze the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHugh-Nagumo oscillators, and determine the ranges of optimal control parameters. We demonstrate that the tweezer control scheme allows for stabilization of chimera states with different shapes, and can be used as an instrument for controlling the coherent domains size, as well as the maximum average frequency difference of the oscillators.
Collapse
Affiliation(s)
- Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | | | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| |
Collapse
|
45
|
Hamilton E, Bruot N, Cicuta P. The chimera state in colloidal phase oscillators with hydrodynamic interaction. CHAOS (WOODBURY, N.Y.) 2017; 27:123108. [PMID: 29289052 DOI: 10.1063/1.4989466] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The chimera state is the incongruous situation where coherent and incoherent populations coexist in sets of identical oscillators. Using driven non-linear oscillators interacting purely through hydrodynamic forces at low Reynolds number, previously studied as a simple model of motile cilia supporting waves, we find concurrent incoherent and synchronised subsets in small arrays. The chimeras seen in simulation display a "breathing" aspect, reminiscent of uniformly interacting phase oscillators. In contrast to other systems where chimera has been observed, this system has a well-defined interaction metric, and we know that the emergent dynamics inherit the symmetry of the underlying Oseen tensor eigenmodes. The chimera state can thus be connected to a superposition of eigenstates, whilst considering the mean interaction strength within and across subsystems allows us to make a connection to more generic (and abstract) chimeras in populations of Kuramoto phase oscillators. From this work, we expect the chimera state to emerge in experimental observations of oscillators coupled through hydrodynamic forces.
Collapse
Affiliation(s)
- Evelyn Hamilton
- Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
| | - Nicolas Bruot
- Institute of Industrial Science, University of Tokyo, Tokyo, Japan
| | - Pietro Cicuta
- Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
| |
Collapse
|
46
|
Bauer L, Bassett J, Hövel P, Kyrychko YN, Blyuss KB. Chimera states in multi-strain epidemic models with temporary immunity. CHAOS (WOODBURY, N.Y.) 2017; 27:114317. [PMID: 29195311 DOI: 10.1063/1.5008386] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibit emergence and annihilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchronization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases.
Collapse
Affiliation(s)
- Larissa Bauer
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Jason Bassett
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Philipp Hövel
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Yuliya N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
| | - Konstantin B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
| |
Collapse
|
47
|
Bick C, Sebek M, Kiss IZ. Robust Weak Chimeras in Oscillator Networks with Delayed Linear and Quadratic Interactions. PHYSICAL REVIEW LETTERS 2017; 119:168301. [PMID: 29099217 DOI: 10.1103/physrevlett.119.168301] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2017] [Indexed: 05/18/2023]
Abstract
We present an approach to generate chimera dynamics (localized frequency synchrony) in oscillator networks with two populations of (at least) two elements using a general method based on a delayed interaction with linear and quadratic terms. The coupling design yields robust chimeras through a phase-model-based design of the delay and the ratio of linear and quadratic components of the interactions. We demonstrate the method in the Brusselator model and experiments with electrochemical oscillators. The technique opens the way to directly bridge chimera dynamics in phase models and real-world oscillator networks.
Collapse
Affiliation(s)
- Christian Bick
- Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, OX2 6GG Oxford , United Kingdom
- Centre for Systems Dynamics and Control and Department of Mathematics, University of Exeter, EX4 4QF Exeter, United Kingdom
| | - Michael Sebek
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
| | - István Z Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St. Louis, Missouri 63103, USA
| |
Collapse
|
48
|
Cho YS, Nishikawa T, Motter AE. Stable Chimeras and Independently Synchronizable Clusters. PHYSICAL REVIEW LETTERS 2017; 119:084101. [PMID: 28952757 DOI: 10.1103/physrevlett.119.084101] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2017] [Indexed: 06/07/2023]
Abstract
Cluster synchronization is a phenomenon in which a network self-organizes into a pattern of synchronized sets. It has been shown that diverse patterns of stable cluster synchronization can be captured by symmetries of the network. Here, we establish a theoretical basis to divide an arbitrary pattern of symmetry clusters into independently synchronizable cluster sets, in which the synchronization stability of the individual clusters in each set is decoupled from that in all the other sets. Using this framework, we suggest a new approach to find permanently stable chimera states by capturing two or more symmetry clusters-at least one stable and one unstable-that compose the entire fully symmetric network.
Collapse
Affiliation(s)
- Young Sul Cho
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
- Department of Physics, Chonbuk National University, Jeonju 561-756, Korea
| | - Takashi Nishikawa
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
- Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, 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
| |
Collapse
|
49
|
Premalatha K, Chandrasekar VK, Senthilvelan M, Lakshmanan M. Chimeralike states in two distinct groups of identical populations of coupled Stuart-Landau oscillators. Phys Rev E 2017; 95:022208. [PMID: 28297891 DOI: 10.1103/physreve.95.022208] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2016] [Indexed: 06/06/2023]
Abstract
We show the existence of chimeralike states in two distinct groups of identical populations of globally coupled Stuart-Landau oscillators. The existence of chimeralike states occurs only for a small range of frequency difference between the two populations, and these states disappear for an increase of mismatch between the frequencies. Here the chimeralike states are characterized by the synchronized oscillations in one population and desynchronized oscillations in another population. We also find that such states observed in two distinct groups of identical populations of nonlocally coupled oscillators are different from the above case in which coexisting domains of synchronized and desynchronized oscillations are observed in one population and the second population exhibits synchronized oscillations for spatially prepared initial conditions. Perturbation from such spatially prepared initial condition leads to the existence of imperfectly synchronized states. An imperfectly synchronized state represents the existence of solitary oscillators which escape from the synchronized group in population I and synchronized oscillations in population II. Also the existence of chimera state is independent of the increase of frequency mismatch between the populations. We also find the coexistence of different dynamical states with respect to different initial conditions, which causes multistability in the globally coupled system. In the case of nonlocal coupling, the system does not show multistability except in the cluster state region.
Collapse
Affiliation(s)
- K Premalatha
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamilnadu, India
| | - M Senthilvelan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| |
Collapse
|
50
|
Maistrenko Y, Brezetsky S, Jaros P, Levchenko R, Kapitaniak T. Smallest chimera states. Phys Rev E 2017; 95:010203. [PMID: 28208439 DOI: 10.1103/physreve.95.010203] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2016] [Indexed: 06/06/2023]
Abstract
We demonstrate that chimera behavior can be observed in small networks consisting of three identical oscillators, with mutual all-to-all coupling. Three different types of chimeras, characterized by the coexistence of two coherent oscillators and one incoherent oscillator (i.e., rotating with another frequency) have been identified, where the oscillators show periodic (two types) and chaotic (one type) behaviors. Typical bifurcations at the transitions from full synchronization to chimera states and between different types of chimeras have been described. Parameter regions for the chimera states are obtained in the form of Arnold tongues, issued from a singular parameter point. Our analysis suggests that chimera states can be observed in small networks relevant to various real-world systems.
Collapse
Affiliation(s)
- Yuri Maistrenko
- Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
- Institute of Mathematics and Centre for Medical and Biotechnical Research, National Academy of Sciences of Ukraine, Tereshchenkivska St. 3, 01030 Kyiv, Ukraine
| | - Serhiy Brezetsky
- Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Patrycja Jaros
- Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Roman Levchenko
- Taras Shevchenko National University of Kyiv, Volodymyrska St. 60, 01030 Kyiv, Ukraine
| | - Tomasz Kapitaniak
- Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| |
Collapse
|