Dynamics and stability of chimera states in two coupled populations of oscillators

Laing, Carlo R.

doi:10.1103/physreve.100.042211

Cited by 24 publications

(14 citation statements)

References 46 publications

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“…Under this condition, functions a(t) and b(t) grow without bounds, making solution (22) unstable and inducing parametric resonance instability in variational equation (18). Expressing Ω, q, β via the original parameters of network (1) and using the first order approximation ω x ≈ Ω p = N −2 N sin α (cf.…”

Section: (B))mentioning

confidence: 99%

“…In the case of identical oscillators, partial synchronization can turn into chimera states that represent fascinating patterns in which even structurally identical oscillators can break into two, possibly asymmetric groups of coherent and incoherent oscillators [16][17][18][19]. Chimera states were extensively studied in the Kuramoto model as well as in other networks of oscillatory systems [19][20][21][22][23][24][25][26][27][28], including coupled chemical oscillators [2], networks of metronomes [29], coupled pendula [30], pedestrians on a bridge [31], optical systems and lasers [32], and continuous media [33,34].…”

Section: Introductionmentioning

confidence: 99%

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Stability of rotatory solitary states in Kuramoto networks with inertia

Munyaev,

Bolotov,

Smirnov

et al. 2021

Preprint

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Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and becomes incoherent with the rest of the network. 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 (i) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (ii) 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.

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Section: (B))mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stability of rotatory solitary states in Kuramoto networks with inertia

Munyaev,

Bolotov,

Smirnov

et al. 2021

Preprint

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“…In the beginning, it was apprehended that the chimera state is a transient behavior; the transient time increases with the size of a network (Wolfrum and Omel'chenko, 2011;Rosin et al, 2014). Later, it has been established that chimera states are possible stable states (Pecora et al, 2014;Omel'chenko, 2018;Laing, 2019) in an ensemble of identical oscillators and with symmetry in the connectivity matrix or the topology of a network. By this time, this phenomenon has been widely explored in single-layer networks (Abrams and Strogatz, 2004;Sethia et al, 2008;Laing, 2009;Hagerstrom et al, 2012;Martens et al, 2013;Omelchenko et al, 2013;Gopal et al, 2014;Hart et al, 2019;Majhi et al, 2019;Parastesh et al, 2020;Wang and Liu, 2020), multilayer networks (Ghosh and Jalan, 2016;Maksimenko et al, 2016;Sawicki et al, 2018;Ruzzene et al, 2020), and 3D networks (Maistrenko et al, 2015;Kasimatis et al, 2018;Kundu et al, 2019) with different forms of chimeras such as traveling chimera (Bera et al, 2016a;Omel'chenko, 2019;Dudkowski et al, 2019;Alvarez-Socorro et al, 2021) and spiral chimera (Martens et al, 2010;Gu et al, 2013).…”

Section: Introductionmentioning

confidence: 99%

Smallest Chimeras Under Repulsive Interactions

Saha¹,

Dana²

2021

Front. Netw. Physiol.

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We present an exemplary system of three identical oscillators in a ring interacting repulsively to show up chimera patterns. The dynamics of individual oscillators is governed by the superconducting Josephson junction. Surprisingly, the repulsive interactions can only establish a symmetry of complete synchrony in the ring, which is broken with increasing repulsive interactions when the junctions pass through serials of asynchronous states (periodic and chaotic) but finally emerge into chimera states. The chimera pattern first appears in chaotic rotational motion of the three junctions when two junctions evolve coherently, while the third junction is incoherent. For larger repulsive coupling, the junctions evolve into another chimera pattern in a periodic state when two junctions remain coherent in rotational motion and one junction transits to incoherent librational motion. This chimera pattern is sensitive to initial conditions in the sense that the chimera state flips to another pattern when two junctions switch to coherent librational motion and the third junction remains in rotational motion, but incoherent. The chimera patterns are detected by using partial and global error functions of the junctions, while the librational and rotational motions are identified by a libration index. All the collective states, complete synchrony, desynchronization, and two chimera patterns are delineated in a parameter plane of the ring of junctions, where the boundaries of complete synchrony are demarcated by using the master stability function.

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“…This simpler structure has been exploited in numerous studies 4,[16][17][18][19][20][21][22][23][24][25][26][27]40 . In many of them the continuum limit was considered 16,18,19 .…”

Section: Introductionmentioning

confidence: 99%

“…Also from a theoretical point of view, an understanding of chimera states plays an important role, as they mediate between order and disorder [13][14][15] . A detailed analysis of their dynamics is much facilitated with a simple topology, the simplest one consisting of two coupled populations [16][17][18][19][20][21][22][23][24][25][26][27] . For this minimal model, analytical results about the stability and bifurcations of chimera states could be obtained in the continuum limit 16 , and for the case of small populations it was shown the same type of bifurcations exist 17 .…”

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confidence: 99%

Attracting Poisson Chimeras in Two-population Networks

Lee,

Krischer

2021

Preprint

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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 ansatz based on the network symmetry-induced cluster pattern dynamics of the finite size network.

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Dynamics and stability of chimera states in two coupled populations of oscillators

Cited by 24 publications

References 46 publications

Stability of rotatory solitary states in Kuramoto networks with inertia

Stability of rotatory solitary states in Kuramoto networks with inertia

Smallest Chimeras Under Repulsive Interactions

Attracting Poisson Chimeras in Two-population Networks

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