Robustness of chimera states in complex dynamical systems

OPEN SUBJECT AREAS: NONLINEAR PHENOMENA PHASE TRANSITIONS AND CRITICAL PHENOMENA Received 6 November 2013 Accepted 29 November 2013 Published 17 December 2013 Correspondence and requests for materials should be addressed to Robustness of chimera states in complex dynamical systems Nan Yao1,2, Zi-Gang Huang2,3, Ying-Cheng Lai2,4 & Zhi-Gang Zheng1 1 Department of Physics and the Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China, 2School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA, 3Lanzhou University, and Institute of Modern Physics of CAS, Lanzhou 730000, China, 4Department of Physics, Arizona State University, Tempe, Arizona 85287, USA. The remarkable phenomenon of chimera state in systems of non-locally coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interests. In such a state, different groups of oscillators can exhibit characteristically distinct types of dynamical behaviors, in spite of identity of the oscillators. But how robust are chimera states against random perturbations to the structure of the underlying network? We address this fundamental issue by studying the effects of random removal of links on the probability for chimera states. Using direct numerical calculations and two independent theoretical approaches, we find that the likelihood of chimera state decreases with the probability of random-link removal. A striking finding is that, even when a large number of links are removed so that chimera states are deemed not possible, in the state space there are generally both coherent and incoherent regions. The regime of chimera state is a particular case in which the oscillators in the coherent region happen to be synchronized or phase-locked. Z.-G.H. (huangzg@ lzu.edu.cn) T he collective dynamics of complex systems are often multifold and much more complicated than the dynamics of individual oscillators. For example, when a large number of oscillators, each possessing very simple dynamics, are coupled together, the collective behaviors of all the oscillators can be highly nontrivial. In the classic Kuramoto network1, each oscillator is coupled with every other oscillator - the configuration of a globally coupled network. Each individual oscillator is a simple rotation of certain frequency, and the dynamics of the oscillators differ only in their frequencies. The coupling function is also a simple mathematical function, such as a sinusoidal type of function. For relatively weak coupling the motions of the oscillators are incoherent, due to the heterogeneity in their frequencies, but as the coupling parameter increases through a critical value, coherence can emerge and persist in the form of partial or complete synchronization1–4. There is now a large body of literature on synchronization in the Kuramoto network, due to its relevance to many physical, chemical, and biological phenomena5. While the emergence of synchronous behavior as the coupling is strengthened is intuitively reasonable and anticipated in any coupled oscillator network, complex systems often present us with unexpected and sometimes quite surprising phenomena. A striking example is the occurrence of chimera state6–24 in non-locally coupled networks of identical oscillators, where different subsets of the oscillators can exhibit completely distinct dynamical behaviors. For example, for a simple form of chimera state, there are two distinct types of behavior among all oscillators in the network: one group of oscillators is nearly perfectly synchronous but the oscillators in the complementary group are completely incoherent. These two types of behaviors emerge as one state of the networked system, in contrast to the phenomenon of multiple coexisting attractors in nonlinear dynamical systems25,26, each with its own basin of attraction. In such a system, while the attractors coexist in the phase space, starting from a single initial condition the system approaches asymptotically to only one attractor of certain characteristics, which can be a stable fixed point, a limit cycle, a quasiperiodic state, or even a chaotic attractor, but from the same initial condition the system cannot simultaneously possess more than one of these traits. Signatures of chimera states were first observed from the spatiotemporal evolution of a system of coupled nonlinear oscillators and the phenomenon was named ‘‘domain-like spatial structure’’27. Chimera states in highly regular and non-locally coupled networks of identical oscillators are thus a quite remarkable type of collective dynamics. We note that nonlocal coupling is relevant to physical systems such as the Josephson-junction arrays28 and to chemical oscillators5,6 as well. The paradigmatic setting in which chimera states have been studied theoretically and computationally is that of non-locally coupled phase oscillators6–19. A fundamental question is how robust chimera states are with respect to SCIENTIFIC REPORTS | 3 : 3522 | DOI: 10.1038/srep03522 1 www.nature.com/scientificreports perturbations. That is, when the system details deviate from those of the paradigmatic setting or when noise is present, can chimera states still emerge and sustain? In this regard, the issue of noise has been successfully addressed, as chimera states have been experimentally observed in a chemical21 and an optical22 systems that are intrinsically noisy. An outstanding issue is then how random perturbations to the network structure affect the chimera states. In this paper, we address this structural robustness issue that is fundamental to our understanding of chimera states. In particular, starting from the standard setting of a non-locally coupled array of identical phase oscillators, we remove links systematically but randomly according to the removal probability p, and investigate whether and to what extent chimera states can persist as p is increased from zero. For a fixed value of p, for an infinite network there are an infinite number of possible configurations. For a realistic network of finite size, the number of configurations can still be extremely large. Due to the randomness in the network structure, the persistence of chimera states can be characterized but in a statistical sense. In particular, given p, certain fraction of the network configurations would permit chimera states, while others would not. One can thus define a probability for chimera states, denoted by F(p), where F(p) R 1 for p R 0 and in general we expect F(p) to be a decreasing function of p. Our extensive computations reveal that chimera states can persist for a range of p values in the sense that F(p) maintains values close to unity even when p is appreciably away from zero, strongly suggesting that the exotic dynamical states are robust with respect to random structural perturbations to the underlying net (...truncated)