Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation

Schneider, Isabelle

doi:10.1098/rsta.2012.0472

Cited by 30 publications

(30 citation statements)

References 12 publications

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“…They could thus be particularly suitable for networks of semiconductor lasers and in neuroscience. Similar extensions of the Pyragas control scheme based on the symmetry-breaking of equivariant Hopf bifurcation [55,56] have recently been suggested and applied to noninvasive and pattern-selective stabilization of three coupled Stuart-Landau oscillators (ponies on a merrygo-round) [55].…”

Section: Introductionmentioning

confidence: 99%

Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

Choe¹,

Kim²,

Jang³

et al. 2014

Int. J. Dynam. Control

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We suggest a delayed feedback control scheme with arbitrary delay for stabilizing a periodic orbit, while maintaining the noninvasiveness of the controller. Since the constraint on the delay to be adjusted to the period of the unstable periodic orbit is not imposed, a richer structure of the dynamics can be observed: Not only weakly unstable, but also strongly unstable periodic orbits are stabilized and even stabilization of orbits with infinite period is achieved. The control mechanism is elucidated for the generic model of a subcritical Hopf bifurcation. A complete bifurcation analysis for the fixed point as well as the periodic orbit is presented and the stability domains are identified. Furthermore, we study the effects of distributed delayed feedback on the stabilization of periodic orbits, and show that larger variance of the delay distribution considerably enlarges the stabilization region in the parameter space. We extend the control scheme to a network of Hopf normal forms coupled with heterogeneous delays. By tuning the coupling parameters, different synchronization patterns, i.e., in-phase, splay, and clustering, can be selected. The characteristic equation for Floquet exponents of the heterogeneous delay network is derived in an analytical form, which reveals the coupling parameters for successful stabilization. The equation takes a C.-U. Choe · R.-S. Kim · H. Jang Center for Nonlinear Science, University of Science, Unjong-District, Pyongyang, DPR Korea P. Hövel · E. Schöll (B) Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany e-mail: schoell@physik.tu-berlin.de P. Hövel Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany unified form for both subcritical and supercritical Hopf bifurcations regardless of the synchronization patterns. Analysis of Floquet exponents and direct numerical simulations show that the heterogeneity in the delays drastically facilitates stabilization and provides an enlarged parameter region for successful control. Finally, we consider the thermodynamic limit in the framework of a mean field approximation and show that heterogeneous delays offer an enhanced performance of control.

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Section: Introductionmentioning

confidence: 99%

Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

Choe¹,

Kim²,

Jang³

et al. 2014

Int. J. Dynam. Control

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“…Coupled nonlinear systems exhibit a plethora of different collective behavior such as partial or cluster synchronization [1][2][3][4][5][6], spatio-temporal patterns under delayed feedback control [7][8][9], and chimera states which consist of spatially coexisting domains of synchronized and unsynchronized dynamics [10][11][12]. Recently, special attention has been paid to the different types of oscillation quenching, amplitude death and oscillation death [13][14][15][16].…”

Section: Introductionmentioning

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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“…We analyze the paradigmatic model of Stuart-Landau (SL) oscillators [24,[43][44][45][46][47][48][49],…”

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confidence: 99%

Chimera Death: Symmetry Breaking in Dynamical Networks

2014

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For a network of generic oscillators with nonlocal topology and symmetry-breaking coupling we establish novel partially coherent inhomogeneous spatial patterns, which combine the features of chimera states (coexisting incongruous coherent and incoherent domains) and oscillation death (oscillation suppression), which we call chimera death. We show that due to the interplay of nonlocality and breaking of rotational symmetry by the coupling two distinct scenarios from oscillatory behavior to a stationary state regime are possible: a transition from an amplitude chimera to chimera death via in-phase synchronized oscillations, and a direct abrupt transition for larger coupling strength. Spontaneous symmetry breaking in a complex dynamical system is a fundamental and universal phenomenon which occurs in diverse fields such as physics, chemistry, and biology [1]. It implies that processes occurring in nature favor a less symmetric configuration, although the underlying principles can be symmetric. This intriguing concept has recently gained renewed interest generated by the enormous burst of works on chimera states and on oscillation death, which have emerged independently. In this Letter we draw a relation between these two.Chimera states correspond to the situation when an ensemble of identical elements self-organizes into two coexisting and spatially separated domains with dramatically different behavior, i.e., spatially coherent and incoherent oscillations [2,3]. They have been the subject of intensive theoretical investigations, e.g. [4][5][6][7][8][9][10][11][12][13][14][15]. Experimental evidence of chimeras has only recently been provided for optical [16], chemical [17], mechanical [18] and electronic [19] systems. These peculiar hybrid states may also account for the observation of partial synchrony in neural activity [20], like unihemispheric sleep, i.e., the ability of some birds or dolphins to sleep with one half of their brain while the other half remains aware [21,22]. Chimera states have been initially found for phase oscillators [2], and they typically occur in networks with nonlocal coupling. Recently, for globally coupled oscillators it has been shown that such spatio-temporal patterns can be also connected to the amplitude dynamics (amplitude-mediated chimeras) [23,24]. However, the global coupling topology does not provide a clear notion of space, which is crucial for chimera states. A further open question is whether chimeras can be extended to more general symmetry breaking states.Another fascinating effect which requires the break-up of the system's symmetry as a crucial ingredient is oscillation death which refers to stable inhomogeneous steady states (IHSS) which are created through the coupling of self-sustained oscillators. This regime occurs when a ho- * corresponding author: schoell@physik.tu-berlin.de mogeneous steady state splits into at least two distinct branches -upper and lower -which represent a newly created IHSS [25][26][27][28]. For a network of coupled elements oscillation death ...

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Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation

Abstract: One contribution of 15 to a Theme Issue 'Dynamics, control and information in delay-coupled systems' .

Cited by 30 publications

References 12 publications

Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

Stable and transient multicluster oscillation death in nonlocally coupled networks

Chimera Death: Symmetry Breaking in Dynamical Networks

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