604 research outputs found
Designing heteroclinic and excitable networks in phase space using two populations of coupled cells
We give a constructive method for realizing an arbitrary directed graph (with
no one-cycles) as a heteroclinic or an excitable dynamic network in the phase
space of a system of coupled cells of two types. In each case, the system is
expressed as a system of first order differential equations. One of the cell
types (the -cells) interacts by mutual inhibition and classifies which
vertex (state) we are currently close to, while the other cell type (the
-cells) excites the -cells selectively and becomes active only when there
is a transition between vertices. We exhibit open sets of parameter values such
that these dynamical networks exist and demonstrate via numerical simulation
that they can be attractors for suitably chosen parameters
Weak chimeras in minimal networks of coupled phase oscillators
We suggest a definition for a type of chimera state that appears in networks
of indistinguishable phase oscillators. Defining a "weak chimera" as a type of
invariant set showing partial frequency synchronization, we show that this
means they cannot appear in phase oscillator networks that are either globally
coupled or too small. We exhibit various networks of four, six and ten
indistinguishable oscillators where weak chimeras exist with various dynamics
and stabilities. We examine the role of Kuramoto-Sakaguchi coupling in giving
degenerate (neutrally stable) families of weak chimera states in these example
networks.Comment: 9 figure
Hopf normal form with symmetry and reduction to systems of nonlinearly coupled phase oscillators
Coupled oscillator models where oscillators are identical and
symmetrically coupled to all others with full permutation symmetry are
found in a variety of applications. Much, but not all, work on phase
descriptions of such systems consider the special case of pairwise coupling
between oscillators. In this paper, we show this is restrictive - and we
characterise generic multi-way interactions between oscillators that are
typically present, except at the very lowest order near a Hopf bifurcation
where the oscillations emerge. We examine a network of identical weakly coupled
dynamical systems that are close to a supercritical Hopf bifurcation by
considering two parameters, (the strength of coupling) and
(an unfolding parameter for the Hopf bifurcation). For small enough
there is an attractor that is the product of stable limit cycles; this
persists as a normally hyperbolic invariant torus for sufficiently small
. Using equivariant normal form theory, we derive a generic normal
form for a system of coupled phase oscillators with symmetry. For fixed
and taking the limit , we show that the
attracting dynamics of the system on the torus can be well approximated by a
coupled phase oscillator system that, to lowest order, is the well-known
Kuramoto-Sakaguchi system of coupled oscillators. The next order of
approximation genericlly includes terms with up to four interacting phases,
regardless of . Using a normalization that maintains nontrivial interactions
in the limit , we show that the additional terms can lead
to new phenomena in terms of coexistence of two-cluster states with the same
phase difference but different cluster size
Dynamics on unbounded domains; co-solutions and inheritance of stability
We consider the dynamics of semiflows of patterns on unbounded domains that
are equivariant under a noncompact group action. We exploit the unbounded
nature of the domain in a setting where there is a strong `global' norm and a
weak `local' norm. Relative equilibria whose group orbits are closed manifolds
for a compact group action need not be closed in a noncompact setting; the
closure of a group orbit of a solution can contain `co-solutions'.
The main result of the paper is to show that co-solutions inherit stability
in the sense that co-solutions of a Lyapunov stable pattern are also stable
(but in a weaker sense). This means that the existence of a single group orbit
of stable relative equilibria may force the existence of quite distinct group
orbits of relative equilibria, and these are also stable. This is in contrast
to the case for finite dimensional dynamical systems where group orbits of
relative equilibria are typically isolated
Multi-cluster dynamics in coupled phase oscillator networks
In this paper we examine robust clustering behaviour with multiple nontrivial
clusters for identically and globally coupled phase oscillators. These systems
are such that the dynamics is completely determined by the number of
oscillators N and a single scalar function (the coupling
function). Previous work has shown that (a) any clustering can stably appear
via choice of a suitable coupling function and (b) open sets of coupling
functions can generate heteroclinic network attractors between cluster states
of saddle type, though there seem to be no examples where saddles with more
than two nontrivial clusters are involved. In this work we clarify the
relationship between the coupling function and the dynamics. We focus on cases
where the clusters are inequivalent in the sense of not being related by a
temporal symmetry, and demonstrate that there are coupling functions that give
robust heteroclinic networks between periodic states involving three or more
nontrivial clusters. We consider an example for N=6 oscillators where the
clustering is into three inequivalent clusters. We also discuss some aspects of
the bifurcation structure for periodic multi-cluster states and show that the
transverse stability of inequivalent clusters can, to a large extent, be varied
independently of the tangential stability
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
Symbolic analysis for some planar piecewise linear maps
In this paper a class of linear maps on the 2-torus and some planar piecewise
isometries are discussed. For these discontinuous maps, by introducing codings
underlying the map operations, symbolic descriptions of the dynamics and
admissibility conditions for itineraries are given, and explicit expressions in
terms of the codings for periodic points are presented.Comment: 4 Figure
Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory
We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motivated by a low-order model for magnetic activity in a stellar dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations, where the saddle-node bifurcation is assumed to have global reinjection of trajectories. The model can produce chaotic behaviour within each of a pair of invariant subspaces, and also it can show attractors that are stuck-on to both of the invariant subspaces. We investigate the detailed intermittent dynamics for such an attractor, investigating the effect of breaking the symmetry between the two Hopf bifurcations, and observing that it can appear via blowout bifurcations from the invariant subspaces.
We give a simple Markov chain model for the two-state intermittent dynamics that reproduces the time spent close to the invariant subspaces and the switching between the different possible invariant subspaces; this clarifies the observation that the proportion of time spent near the different subspaces depends on the average residence time and also on the probabilities of switching between the possible subspaces
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