Heteroclinic Ratchets in a System of Four Coupled Oscillators

We study an unusual but robust phenomenon that appears in an example system of four coupled phase oscillators. We show that the system can have a robust attractor that responds to a specific detuning between certain pairs of the oscillators by a breaking of phase locking for arbitrary positive detunings but not for negative detunings. As the dynamical mechanism behind this is a particular type of heteroclinic network, we call this a 'heteroclinic ratchet' because of its dynamical resemblance to a mechanical ratchet

Convergence of Time Averages Near Statistical Attractors and Ratcheting of Coupled Oscillators

In this thesis, convergence of time averages near statistical attractors of continuous flows are investigated. A relation between statistical attractor and essential $\omega$-limit set is proved, and using this a general definition for statistical attractor is given. Sufficient conditions are given for an observable to admit a convergent time average along the orbits of the flow. The general results are applied to flows on a torus, and in particular to systems of coupled phase oscillators that admit attracting heteroclinic networks in their phase space. A particular heteroclinic network that we call heteroclinic ratchet is observed and analysed in detail. Heteroclinic ratchets give rise to a novel phenomenon, unidirectional desynchronization of oscillators (ratcheting). The results obtained about the convergence of time averages near statistical attractors implies that heteroclinic ratchets induce, besides its other interesting consequences, frequency synchronization without phase synchronization. Different coupling structures that can give rise to ratcheting of oscillators are also investigated

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

On designing heteroclinic networks from graphs

Copyright © 2013 Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in Physica D: Nonlinear Phenomena. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica D: Nonlinear Phenomena Vol. 265 (2013), DOI: 10.1016/j.physd.2013.09.006Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a very complicated structure that is poorly understood and determined to a large extent by the constraints and dimension of the system. As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand how particular graphs can be realised as robust heteroclinic networks that are attracting. This paper presents two methods of realizing arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs---we say the ODEs {\em realise} the graphs as heteroclinic networks between equilibria that represent the vertices. Suppose we have a directed graph on $n_v$ vertices with $n_e$ edges. The "simplex realisation" embeds the graph as an invariant set of a flow on an $(n_v-1)$-simplex. This method realises the graph as long as it is one- and two-cycle free. The "cylinder realisation" embeds a graph as an invariant set of a flow on a $(n_e+1)$-dimensional space. This method realises the graph as long as it is one-cycle free. In both cases we find the graph as an invariant set within an attractor, and discuss some illustrative examples, including the influence of noise and parameters on the dynamics. In particular we show that the resulting heteroclinic network may or may not display "memory" of the vertices visited.Mathematical Biosciences Institute (MBI), OhioRoyal SocietyUniversity of Aucklan

Criteria for robustness of heteroclinic cycles in neural microcircuits.

Copyright © 2011 Ashwin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.We introduce a test for robustness of heteroclinic cycles that appear in neural microcircuits modeled as coupled dynamical cells. Robust heteroclinic cycles (RHCs) can appear as robust attractors in Lotka-Volterra-type winnerless competition (WLC) models as well as in more general coupled and/or symmetric systems. It has been previously suggested that RHCs may be relevant to a range of neural activities, from encoding and binding to spatio-temporal sequence generation.The robustness or otherwise of such cycles depends both on the coupling structure and the internal structure of the neurons. We verify that robust heteroclinic cycles can appear in systems of three identical cells, but only if we require perturbations to preserve some invariant subspaces for the individual cells. On the other hand, heteroclinic attractors can appear robustly in systems of four or more identical cells for some symmetric coupling patterns, without restriction on the internal dynamics of the cells

On statistical attractors and the convergence of time averages

Copyright © 2011 Cambridge Philosophical SocietyThere are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems

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

From coupled networks of systems to networks of states in phase space

This is the author accepted manuscript. The final version is available from American Institute of Mathematical Sciences (AIMS) via the DOI in this record.Dynamical systems on graphs can show a wide range of behaviours beyond simple synchronization - even simple globally coupled structures can exhibit attractors with intermittent and slow switching between patterns of synchrony. Such attractors, called heteroclinic networks, can be well described as networks in phase space and in this paper we review some results and examples of how these robust attractors can be characterised from the synchrony properties as well how coupled systems can be designed to exhibit given but arbitrary network attractors in phase space

Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation

Copyright © 2011 Springer. The final publication is available at www.springerlink.comWe consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible “inflations” of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells