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 $p$-cells) interacts by mutual inhibition and classifies which vertex (state) we are currently close to, while the other cell type (the $y$-cells) excites the $p$-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 SNS_N symmetry and reduction to systems of nonlinearly coupled phase oscillators

Coupled oscillator models where $N$ oscillators are identical and symmetrically coupled to all others with full permutation symmetry $S_N$ 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, $\epsilon$ (the strength of coupling) and $\lambda$ (an unfolding parameter for the Hopf bifurcation). For small enough $\lambda>0$ there is an attractor that is the product of $N$ stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small $\epsilon>0$. Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with $S_N$ symmetry. For fixed $N$ and taking the limit $0<\epsilon\ll\lambda\ll 1$, 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 $N$. Using a normalization that maintains nontrivial interactions in the limit $N\rightarrow \infty$, 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

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 $g(\varphi)$ (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

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

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