27 research outputs found
Isotropy of Angular Frequencies and Weak Chimeras With Broken Symmetry
The notion of a weak chimeras provides a tractable definition for chimera
states in networks of finitely many phase oscillators. Here we generalize the
definition of a weak chimera to a more general class of equivariant dynamical
systems by characterizing solutions in terms of the isotropy of their angular
frequency vector - for coupled phase oscillators the angular frequency vector
is given by the average of the vector field along a trajectory. Symmetries of
solutions automatically imply angular frequency synchronization. We show that
the presence of such symmetries is not necessary by giving a result for the
existence of weak chimeras without instantaneous or setwise symmetries for
coupled phase oscillators. Moreover, we construct a coupling function that
gives rise to chaotic weak chimeras without symmetry in weakly coupled
populations of phase oscillators with generalized coupling
Chaotic Weak Chimeras and their Persistence in Coupled Populations of Phase Oscillators
Nontrivial collective behavior may emerge from the interactive dynamics of
many oscillatory units. Chimera states are chaotic patterns of spatially
localized coherent and incoherent oscillations. The recently-introduced notion
of a weak chimera gives a rigorously testable characterization of chimera
states for finite-dimensional phase oscillator networks. In this paper we give
some persistence results for dynamically invariant sets under perturbations and
apply them to coupled populations of phase oscillators with generalized
coupling. In contrast to the weak chimeras with nonpositive maximal Lyapunov
exponents constructed so far, we show that weak chimeras that are chaotic can
exist in the limit of vanishing coupling between coupled populations of phase
oscillators. We present numerical evidence that positive Lyapunov exponents can
persist for a positive measure set of this inter-population coupling strength
Chaos in generically coupled phase oscillator networks with nonpairwise interactions
The Kuramoto-Sakaguchi system of coupled phase oscillators, where interaction
between oscillators is determined by a single harmonic of phase differences of
pairs of oscillators, has very simple emergent dynamics in the case of
identical oscillators that are globally coupled: there is a variational
structure that means the only attractors are full synchrony (in-phase) or splay
phase (rotating wave/full asynchrony) oscillations and the bifurcation between
these states is highly degenerate. Here we show that nonpairwise coupling -
including three and four-way interactions of the oscillator phases - that
appears generically at the next order in normal-form based calculations, can
give rise to complex emergent dynamics in symmetric phase oscillator networks.
In particular, we show that chaos can appear in the smallest possible dimension
of four coupled phase oscillators for a range of parameter values
Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators
Despite their simplicity, networks of coupled phase oscillators can give rise
to intriguing collective dynamical phenomena. However, the symmetries of
globally and identically coupled identical units do not allow solutions where
distinct oscillators are frequency-unlocked -- a necessary condition for the
emergence of chimeras. Thus, forced symmetry breaking is necessary to observe
chimera-type solutions. Here, we consider the bifurcations that arise when full
permutational symmetry is broken for the network to consist of coupled
populations. We consider the smallest possible network composed of four phase
oscillators and elucidate the phase space structure, (partial) integrability
for some parameter values, and how the bifurcations away from full symmetry
lead to frequency-unlocked weak chimera solutions. Since such solutions wind
around a torus they must arise in a global bifurcation scenario. Moreover,
periodic weak chimeras undergo a period doubling cascade leading to chaos. The
resulting chaotic dynamics with distinct frequencies do not rely on amplitude
variation and arise in the smallest networks that support chaos
Dynamics and Synchronization of Weak Chimera States for a Coupled Oscillator System
This thesis is an investigation of chimera states in a network of identical coupled phase
oscillators. Chimera states are intriguing phenomena that can occur in systems of coupled
identical phase oscillators when synchronized and desynchronized oscillators coexist.
We use the Kuramoto model and coupling function of Hansel for a specific system of six
oscillators to prove the existence of chimera states.
More precisely, we prove analytically there are chimera states in a small network of
six phase oscillators previously investigated numerically by Ashwin and Burylko [8]. We
can reduce to a two-dimensional system within an invariant subspace, in terms of phase
differences. This system is found to have an integral of motion for a specific choice of
parameters. Using this we prove there is a set of periodic orbits that is a weak chimera.
Moreover, we are able to confirmthat there is an infinite number of chimera states at the
special case of parameters, using the weak chimera definition of [8].
We approximate the Poincaré return map for these weak chimera solutions and demonstrate
several results about their stability and bifurcation for nearby parameters. These agree
with numerical path following of the solutions.
We also consider another invariant subspace to reduce the Kuramoto model of six
coupled phase oscillators to a first order differential equation. We analyse this equation
numerically and find regions of attracting chimera states exist within this invariant subspace.
By computing eigenvalues at a nonhyperbolic point for the system of phase differences, we
numerically find there are chimera states in the invariant subspace that are attracting within
full system.Republic of Iraq,
Ministry of Higher Education and Scientific Research
Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks
In the first part of this paper, we showed that three coupled populations of
identical phase oscillators give rise to heteroclinic cycles between invariant
sets where populations show distinct frequencies. Here, we now give explicit
stability results for these heteroclinic cycles for populations consisting of
two oscillators each. In systems with four coupled phase oscillator
populations, different heteroclinic cycles can form a heteroclinic network.
While such networks cannot be asymptotically stable, the local attraction
properties of each cycle in the network can be quantified by stability indices.
We calculate these stability indices in terms of the coupling parameters
between oscillator populations. Hence, our results elucidate how oscillator
coupling influences sequential transitions along a heteroclinic network where
individual oscillator populations switch sequentially between a high and a low
frequency regime; such dynamics appear relevant for the functionality of neural
oscillators