Cascades of Multiheaded Chimera States for Coupled Phase Oscillators

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

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Mathematical Framework for Breathing Chimera States

Journal of Nonlinear Science ◽

10.1007/s00332-021-09779-1 ◽

2022 ◽

Vol 32 (2) ◽

Author(s):

O. E. Omel’chenko

Keyword(s):

Coupled Oscillators ◽

Analytic Solution ◽

Coarse Grained ◽

Mathematical Framework ◽

Intensive Study ◽

Phase Oscillators ◽

Chimera States ◽

The Subject ◽

Self Consistency

AbstractAbout two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence–incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott–Antonsen reduction technique.

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CHIMERA STATES IN A RING OF NONLOCALLY COUPLED OSCILLATORS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127406014551 ◽

2006 ◽

Vol 16 (01) ◽

pp. 21-37 ◽

Cited By ~ 161

Author(s):

DANIEL M. ABRAMS ◽

STEVEN H. STROGATZ

Keyword(s):

Traveling Waves ◽

Collective Behavior ◽

Coupled Oscillators ◽

Spatiotemporal Chaos ◽

Phase Oscillators ◽

Nonlocal Coupling ◽

One Dimensional ◽

Chimera State ◽

Global Coupling ◽

Pattern Forming

Arrays of identical limit-cycle oscillators have been used to model a wide variety of pattern-forming systems, such as neural networks, convecting fluids, laser arrays and coupled biochemical oscillators. These systems are known to exhibit rich collective behavior, from synchrony and traveling waves to spatiotemporal chaos and incoherence. Recently, Kuramoto and his colleagues reported a strange new mode of organization — here called the chimera state — in which coherence and incoherence exist side by side in the same system of oscillators. Such states have never been seen in systems with either local or global coupling; they are apparently peculiar to the intermediate case of nonlocal coupling. Here we give an exact solution for the chimera state, for a one-dimensional ring of phase oscillators coupled nonlocally by a cosine kernel. The analysis reveals that the chimera is born in a continuous bifurcation from a spatially modulated drift state, and dies in a saddle-node collision with an unstable version of itself.

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Emergence of Stripe-Core Mixed Spiral Chimera on a Spherical Surface of Nonlocally Coupled Oscillators

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421501820 ◽

2021 ◽

Vol 31 (12) ◽

pp. 2150182

Author(s):

Ryong-Son Kim ◽

Gi-Hun Tae ◽

Chol-Ung Choe

Keyword(s):

Coupled Oscillators ◽

Spherical Surface ◽

Spiral Wave ◽

Stability Diagram ◽

Phase Oscillators ◽

Type Region ◽

Chimera State ◽

Existence And Stability ◽

The Stability ◽

Theoretical Results

We report on a stripe-core mixed spiral chimera in a system of nonlocally coupled phase oscillators, located on the spherical surface, where the spiral wave consisting of phase-locked oscillators is separated by a stripe-type region of incoherent oscillators into two parts. We analyze the existence and stability of the stripe-core mixed spiral chimera state rigorously, on the basis of the Ott–Antonsen reduction theory. The stability diagram for the stationary states including the spiral chimeras as well as incoherent state is presented. Our stability analysis reveals that the stripe-core mixed spiral chimera state emerges as a unique attractor and loses its stability via the Hopf bifurcation. We verify our theoretical results using direct numerical simulations of the model system.

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The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues

Abstract and Applied Analysis ◽

10.1155/2013/152518 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11

Author(s):

Xiaodong Li ◽

Weipeng Zhang ◽

Fengjie Geng ◽

Jicai Huang

Keyword(s):

Periodic Orbit ◽

Homoclinic Orbit ◽

Dimensional System ◽

Bifurcation Diagrams ◽

Saddle Node Bifurcation ◽

Saddle Node ◽

Higher Dimensional

The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

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RESONANCE STRUCTURE IN A WEAKLY DETUNED LASER WITH INJECTED SIGNAL

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401003693 ◽

2001 ◽

Vol 11 (10) ◽

pp. 2587-2605 ◽

Cited By ~ 10

Author(s):

CATALINA MAYOL ◽

MARIO A. NATIELLO ◽

MARTÍN G. ZIMMERMANN

Keyword(s):

Hopf Bifurcation ◽

Periodic Orbit ◽

Global Bifurcation ◽

Homoclinic Orbits ◽

Resonance Structure ◽

Saddle Node Bifurcation ◽

Arnold Tongue ◽

Saddle Node ◽

Organizing Center ◽

Qualitative Dynamics

We describe the qualitative dynamics and bifurcation set for a laser with injected signal for small cavity detunings. The main organizing center is the Hopf-saddle-node bifurcation from where a secondary Hopf bifurcation of a periodic orbit originates. We show that the laser's stable cw solution existing for low injections, also suffers a secondary Hopf bifurcation. The resonance structure of both tori interact, and homoclinic orbits to the "off" state are found inside each Arnold tongue. The accumulation of all the above resonances towards the Hopf-saddle-node singularity points to the occurrence of a highly degenerate global bifurcation at the codimension-2 point.

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The Shape of Phase-Resetting Curves in Oscillators with a Saddle Node on an Invariant Circle Bifurcation

Neural Computation ◽

10.1162/neco_a_00370 ◽

2012 ◽

Vol 24 (12) ◽

pp. 3111-3125 ◽

Cited By ~ 24

Author(s):

G. Bard Ermentrout ◽

Leon Glass ◽

Bart E. Oldeman

Keyword(s):

Limit Cycle ◽

Two Dimensional ◽

Phase Resetting ◽

Dimensional Model ◽

Bifurcation Structure ◽

Invariant Circle ◽

Saddle Node Bifurcation ◽

Saddle Node ◽

Two Dimensional Model ◽

Parameter Values

We introduce a simple two-dimensional model that extends the Poincaré oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle (SNIC) for certain parameter values. Arbitrarily close to this bifurcation, the phase-resetting curve (PRC) continuously depends on parameters, where its shape can be not only primarily positive or primarily negative but also nearly sinusoidal. This example system shows that one must be careful inferring anything about the bifurcation structure of the oscillator from the shape of its PRC.

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DEGENERATE BIFURCATION SCENARIOS IN THE DYNAMICS OF COUPLED OSCILLATORS WITH SYMMETRIC NONLINEARITIES

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740601468x ◽

2006 ◽

Vol 16 (01) ◽

pp. 169-178 ◽

Cited By ~ 2

Author(s):

O. V. GENDELMAN

Keyword(s):

Coupled Oscillators ◽

Coupling Parameter ◽

Normal Modes ◽

Nonlinear Normal Modes ◽

Saddle Node Bifurcation ◽

High Energies ◽

Saddle Node ◽

Weakly Coupled ◽

Nonlinear Normal ◽

Infinite Energy

We study the degenerate bifurcations of nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. The potentials of both the oscillator and the coupling spring are adopted to be even-power polynomials with non-negative coefficients. Coupling parameter ε is defined and the bifurcations of the nonlinear normal modes structure with change of this coupling parameter are revealed. The degeneracy in the dynamics is manifested by a "bifurcation from infinity" where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Other (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1 : 1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by the mechanism of successive cusp catastrophes with the growth of coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincaré sections). In the particular case of pure cubic nonlinearity of the oscillator and the coupling spring, an agreement between quantitative analytical predictions and numerical results is observed.

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Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415501850 ◽

2015 ◽

Vol 25 (13) ◽

pp. 1550185 ◽

Cited By ~ 3

Author(s):

Tetsushi Ueta ◽

Daisuke Ito ◽

Kazuyuki Aihara

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Stable Limit Cycle ◽

Period Doubling ◽

Transient Behavior ◽

Control Input ◽

Stable Limit ◽

Unstable Periodic Orbits ◽

Saddle Node Bifurcation ◽

Saddle Node

We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark–Sacker bifurcations whose periodic orbits continue to exist through the bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate the suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic orbit can restore a stable limit cycle which disappeared after the saddle-node bifurcation.

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