Symmetry-breaking transitions in networks of nonlinear circuit elements

Heinrich, Martin; Dahms, Thomas; Flunkert, Valentín; Teitsworth, Stephen W.; Schöll, Eckehard

doi:10.1088/1367-2630/12/11/113030

Cited by 75 publications

(35 citation statements)

References 61 publications

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“…Therefore, amplitude death is represented by a symmetric homogeneous steady state, whereas oscillation death is characterized by an inhomogeneous steady state. Oscillation death associated with the stabilization of inhomogeneous steady states has been shown to exist in various systems, e.g., tunnel diode circuits [31], electrochemical [32], chemical droplets [33] and biological systems like neuronal networks [34], calcium oscillators [35], genetic oscillators [36], stem cell differentiation [37]. Oscillation death is especially relevant for biological systems, since it provides a mechanism for cellular differentiation.…”

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confidence: 99%

Time delay control of symmetry-breaking primary and secondary oscillation death

Zakharova¹,

Schneider²,

Kyrychko³

et al. 2013

EPL

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-We show that oscillation death as a specific type of oscillation suppression, which implies symmetry breaking, can be controlled by introducing time-delayed coupling. In particular, we demonstrate that time delay influences the stability of an inhomogeneous steady state, providing the opportunity to modulate the threshold for oscillation death. Additionally, we find a novel type of oscillation death representing a secondary bifurcation of an inhomogeneous steady state.Introduction. -Time-delayed couplings arise naturally in many types of networks, for instance in coupled lasers [1], neural networks [2-4], electronic circuits [5], or genetic oscillators [6], due to finite signal transmission and processing times, and memory and latency effects. While investigating real-world systems, it is necessary to take time delay into account, since the presence of time delay is an inherent property of the vast majority of processes that occur in nature [7,8]. Moreover, time-delayed coupling and feedback represent an important aspect of control [9]. Previous theoretical and experimental works have shown that time delay can be treated as a control parameter and can stabilize initially unstable states. In particular, time-delayed feedback has been used to stabilize unstable periodic orbits embedded in a deterministic chaotic attractor [10,11], or generated by a Hopf bifurcation [12], unstable steady states [13], spatio-temporal patterns [14][15][16], or control the coherence and timescales of stochastic motion [17]. In coupled nonlinear systems and networks time-delayed couplings represent an ubiquitous feature [18] which can also be used to control stability.

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confidence: 99%

Time delay control of symmetry-breaking primary and secondary oscillation death

Zakharova¹,

Schneider²,

Kyrychko³

et al. 2013

EPL

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“…Oscillation death has been observed experimentally in many different systems, such as chemical reactors [21], chemical oscillators [22], chemical droplets [23], electronic circuits [24], or thermokinetic oscillators [25]. There exist various biological applications, e.g.…”

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confidence: 99%

Stable and transient multicluster oscillation death in nonlocally coupled networks

et al. 2015

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In a network of nonlocally coupled Stuart-Landau oscillators with symmetry-breaking coupling, we study numerically, and explain analytically, a family of inhomogeneous steady states (oscillation death). They exhibit multi-cluster patterns, depending on the cluster distribution prescribed by the initial conditions. Besides stable oscillation death, we also find a regime of long transients asymptotically approaching synchronized oscillations. To explain these phenomena analytically in dependence on the coupling range and the coupling strength, we first use a mean-field approximation which works well for large coupling ranges but fails for coupling ranges which are small compared to the cluster size. Going beyond standard mean-field theory, we predict the boundaries of the different stability regimes as well as the transient times analytically in excellent agreement with numerical results.

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“…This strongly suggests their universal appearance. Our findings also show that current knowledge of these hybrid states is far from being complete: Next to the classical chimera state, which exhibits one coherent, phase-locked and one incoherent region, we find a new class of dynamics that possesses multiple domains of incoherence.We consider a ring of N nonlocally coupled FitzHughNagumo (FHN) oscillators, whose relevance is not limited to neuroscience, but also includes chemical [20] and optoelectronic [21] oscillators and nonlinear electronic circuits [22]:where u k and v k are the activator and inhibitor variables, respectively [23,24], and ε > 0 is a small parameter characterizing a timescale separation, which we fix at ε = 0.05 throughout the paper. Depending upon the…”

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confidence: 99%

When Nonlocal Coupling between Oscillators Becomes Stronger: Patched Synchrony or Multichimera States

Omelchenko

Omel’chenko²,

Hövel³

et al. 2013

Phys. Rev. Lett.

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410

427

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Systems of nonlocally coupled oscillators can exhibit complex spatio-temporal patterns, called chimera states, which consist of coexisting domains of spatially coherent (synchronized) and incoherent dynamics. We report on a novel form of these states, found in a widely used model of a limit-cycle oscillator if one goes beyond the limit of weak coupling typical for phase oscillators. Then patches of synchronized dynamics appear within the incoherent domain giving rise to a multichimera state. We find that, depending on the coupling strength and range, different multi-chimeras arise in a transition from classical chimera states. The additional spatial modulation is due to strong coupling interaction and thus cannot be observed in simple phase-oscillator models. During recent times the investigation of coupled systems has led to joint research efforts bridging between diverse fields such as nonlinear dynamics, network science, and statistical physics, with a plethora of applications, e.g., in physics, biology, and technology. As the numerical resources have developed at fast pace, analysis and simulations of large networks with more and more sophisticated coupling schemes have come into reach giving rise to an abundance of new dynamical scenarios. Among these a very peculiar type of dynamics was first reported for the well-known model of phase oscillators. Such a network exhibits a hybrid nature combining both coherent and incoherent parts [1][2][3][4], hence the name chimera states. The most surprising aspect of this discovery was that these states exist in a system of identical oscillators coupled in a symmetric ring topology with a symmetric interaction function. Recent works have shown that chimeras are not limited to phase oscillators, but can in fact be found in a large variety of different systems. These include time-discrete and time-continuous chaotic models [5,6] and are not restricted to one spatial dimension. Also two-dimensional configurations allow for chimera states [7,8]. Furthermore, similar scenarios exist for time-delayed coupling [9], and their dynamical properties and symmetries were subject to theoretical studies as well [6,10,11]. It was only in the very recent past that chimeras were realized in experiments on chemical oscillators [12] and electro-optical coupled-map lattices [13]. The nonlocality of the coupling -a crucial ingredient for chimera states -also suggests an interesting connection to material science, see, for instance, magnetic Janus particles that undergo a synchronization-induced structural transition in a rotating magnetic field [14,15]. Nonlo- * corresponding author: schoell@physik.tu-berlin.de cality is of great importance not only for chimera states, but also for other dynamical phenomena such as turbulent intermittency [16]. Hybrid states were also reported in the context of neuroscience under the notion of bump states [17]. They were later confirmed for nonlocally coupled Hodgkin-Huxley models [18] and may account for experimental observation of partial synchrony in neu...

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Symmetry-breaking transitions in networks of nonlinear circuit elements

Cited by 75 publications

References 61 publications

Time delay control of symmetry-breaking primary and secondary oscillation death

Time delay control of symmetry-breaking primary and secondary oscillation death

Stable and transient multicluster oscillation death in nonlocally coupled networks

When Nonlocal Coupling between Oscillators Becomes Stronger: Patched Synchrony or Multichimera States

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