Basin stability for chimera states

Rakshit, Sarbendu; Bera, Bidesh K.; Perc, Matjaž; Ghosh, Dibakar

doi:10.1038/s41598-017-02409-5

Cited by 86 publications

(32 citation statements)

References 57 publications

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“…From the discovery of the chimera states, the initial conditions play a crucial role. For many coupled systems, chimera states emerge only for a few specific initial conditions [57,58]. Especially in multi-stable systems, the emergence of chimera states depends on the initial conditions.…”

Section: Global Stability Of the Chimera And Coherent Statesmentioning

confidence: 99%

“…Especially in multi-stable systems, the emergence of chimera states depends on the initial conditions. Since the isolated Leech neuronal model itself possesses multi-stable behavior, we now investigate the robustness of the initial conditions on the chimera states using basin stability measurement [58,59]. This is a non-linear, non-local approach, that can be easily applied to high-dimensional multi-stable systems.…”

Section: Global Stability Of the Chimera And Coherent Statesmentioning

confidence: 99%

“…The basin stability tells us how stable the incoherent, chimera, and coherent states are against multifarious initial conditions. Recently, this measurement was used to quantify the multi-stable incoherent, chimera, and coherent states in non-locally coupled time-delayed Mackey-Glass oscillators [58]. Here we have characterized different dynamical states by calculating the strength of incoherence.…”

Section: Global Stability Of the Chimera And Coherent Statesmentioning

confidence: 99%

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Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

et al. 2019

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Chimera states have been a vibrant subject of research in the recent past, but the analytical treatment of transitions from chimeras to coherent states remains a challenge. Here we analytically derive the necessary conditions for this transition by means of the coherent stability function approach, which is akin to the master stability function approach that is traditionally used to study the stability of synchronization in coupled oscillators. In chimera states, there is typically at least one group of oscillators that evolves in a drifting, random manner, while other groups of oscillators follow a smoother, more coherent profile. In the coherent state, there thus exists a smooth functional relationship between the oscillators. This lays the foundation for the coherent stability function approach, where we determine the stability of the coherent state. We subsequently test the analytical prediction numerically by calculating the strength of incoherence during the transition point. We use leech neurons, which exhibit a coexistence of chaotic and periodic tonic spiking depending on initial conditions, coupled via non-local electrical synapses, to demonstrate our approach. We systematically explore various dynamical states with the focus on the transitions between chimeras and coherence, fully confirming the validity of the coherent stability function. We also observe complete synchronization for higher values of the coupling strength, which we verify by the master stability function approach.

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Section: Global Stability Of the Chimera And Coherent Statesmentioning

confidence: 99%

Section: Global Stability Of the Chimera And Coherent Statesmentioning

confidence: 99%

Section: Global Stability Of the Chimera And Coherent Statesmentioning

confidence: 99%

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Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

et al. 2019

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“…Recently, the concept of basin stability (BS) was suggested to describe the stability against large perturbations [15,[18][19][20]. The BS method was later used in various applications [21][22][23][24] and also confirmed experimentally [25]. BS is a probabilistic measure that characterizes the likelihood that the perturbation is 'safe', i.e.…”

Section: Introductionmentioning

confidence: 99%

Interval stability for complex systems

et al. 2018

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Stability of dynamical systems against strong perturbations is an important problem of nonlinear dynamics relevant to many applications in various areas. Here, we develop a novel concept of interval stability, referring to the behavior of the perturbed system during a finite time interval. Based on this concept, we suggest new measures of stability, namely interval basin stability (IBS) and interval stability threshold (IST). IBS characterizes the likelihood that the perturbed system returns to the stable regime (attractor) in a given time. IST provides the minimal magnitude of the perturbation capable to disrupt the stable regime for a given interval of time. The suggested measures provide important information about the system susceptibility to external perturbations which may be useful for practical applications. Moreover, from a theoretical viewpoint the interval stability measures are shown to bridge the gap between linear and asymptotic stability. We also suggest numerical algorithms for quantification of the interval stability characteristics and demonstrate their potential for several dynamical systems of various nature, such as power grids and neural networks.

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“…Recent studies have also reported emergence of chimera states in multiplex networks 26 . The effect of initial conditions on chimera states recently investigated using a basin stability approach 27 . We raise one important question here: how to control chimera pattern [28][29][30][31] , mainly, how to restrict the pattern to a targeted population or to a selected set of oscillators?…”

Section: Introductionmentioning

confidence: 99%

Coexisting synchronous and asynchronous states in locally coupled array of oscillators by partial self-feedback control

Bera

Ghosh

Parmananda

et al. 2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We report the emergence of coexisting synchronous and asynchronous subpopulations of oscillators in one dimensional arrays of identical oscillators by applying a self-feedback control. When a self-feedback is applied to a subpopulation of the array, similar to chimera states, it splits into two/more sub-subpopulations coexisting in coherent and incoherent states for a range of self-feedback strength. By tuning the coupling between the nearest neighbors and the amount of self-feedback in the perturbed subpopulation, the size of the coherent and the incoherent sub-subpopulations in the array can be controlled, although the exact size of them is unpredictable. We present numerical evidence using the Landau-Stuart (LS) system and the KuramotoSakaguchi (KS) phase model. The symmetry breaking of a coherent population of identical oscillators under non-local, local and global coupling, into coexisting coherent and incoherent subpopulations is a strange behavior that has been discussed in recent literature of collective behavior of many dynamical systems. Such a self-organized behavior of a homogeneous population of oscillators is called chimera states. We observe emergence of a similar type of splitting into subsets of coexisting synchronous and asynchronous oscillators, when we apply a self-feedback to a coherent subset of oscillators in a locally coupled open chain of identical oscillators. Interestingly, the unperturbed subpopulation remains coherent for higher values of feedback strength. Most encouragingly, it shows a direction how the size of the coherent and the incoherent subpopulations in the chimera pattern can be monitored by playing with the feedback strength. We provide numerical evidence in two such arrays using the Landau-Stuart (LS) system and the Kuramoto-Sakaguchi (KS) phase model.

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Basin stability for chimera states

Cited by 86 publications

References 57 publications

Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

Interval stability for complex systems

Coexisting synchronous and asynchronous states in locally coupled array of oscillators by partial self-feedback control

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