Robust Heteroclinic Networks Research Articles

Finite switching near heteroclinic networks

We address the level of complexity that can be observed in the dynamics near a robust heteroclinic network. We show that infinite switching, which is a path towards chaos, does not exist near a heteroclinic network such that the eigenvalues of the Jacobian matrix at each node are all real. Furthermore, for a path starting at a node that belongs to more than one heteroclinic cycle, we find a bound for the number of such nodes that can exist in any such path. This constricted dynamics is in stark contrast with examples in the literature of heteroclinic networks such that the eigenvalues of the Jacobian matrix at one node are complex.

Bifurcations from an attracting heteroclinic cycle under periodic forcing

There are few examples of non-autonomous vector fields exhibiting complex dynamics that may be proven analytically. We analyse a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two-sphere. We derive the first return map near the heteroclinic cycle for small amplitude of the perturbing term, and we reduce the analysis of the non-autonomous system to that of a two-dimensional map on a cylinder.Interesting dynamical features arise from a discrete-time Bogdanov-Takens bifurcation. When the perturbation strength is small the first return map has an attracting invariant closed curve that is not contractible on the cylinder. Near the centre of frequency locking there are parameter values with bistability: the invariant curve coexists with an attracting fixed point. Increasing the perturbation strength there are periodic solutions that bifurcate into a closed contractible invariant curve and into a region where the dynamics is conjugate to a full shift on two symbols.

Asymptotic stability of robust heteroclinic networks

We provide conditions guaranteeing that certain classes of robust heteroclinic networks are asymptotically stable.We study the asymptotic stability of ac-networks—robust heteroclinic networks that exist in smooth -equivariant dynamical systems defined in the positive orthant of . Generators of the group are the transformations that change the sign of one of the spatial coordinates. The ac-network is a union of hyperbolic equilibria and connecting trajectories, where all equilibria belong to the coordinate axes (not more than one equilibrium per axis) with unstable manifolds of dimension one or two. The classification of ac-networks is carried out by describing all possible types of associated graphs.We prove sufficient conditions for asymptotic stability of ac-networks. The proof is given as a series of theorems and lemmas that are applicable to the ac-networks and to more general types of networks. Finally, we apply these results to discuss the asymptotic stability of several examples of heteroclinic networks.

Almost Complete and Equable Heteroclinic Networks

Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the relation between the heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. We consider realizations of a large class of transitive digraphs as robust heteroclinic networks and show that although robust realizations are typically not complete (i.e. not all unstable manifolds of nodes are part of the network), they can be almost complete (i.e. complete up to a set of zero measure within the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). We show there are almost complete and equable realizations that can be closed by adding a number of extra nodes and connections. We discuss some examples and describe a sense in which an equable almost complete network embedding is an optimal description of stochastically perturbed motion on the network.

Patterns of desynchronization and resynchronization in heteroclinic networks

We prove results that enable the efficient and natural realization of a large class of robust heteroclinic networks in coupled identical cell systems. We also propose some general conjectures that relate a natural and large class of robust heteroclinic networks that occur in networks modelled by equations of Lotka–Volterra type, and certain networks of symmetric systems, to robust heteroclinic networks in coupled cell networks.

Heteroclinic network dynamics on joining coupled cell networks

ABSTRACTWe present a method of combining coupled cell systems to get dynamics supporting robust simple heteroclinic networks given by the product of robust simple heteroclinic networks (cycles). We consider coupled cell networks, with no assumption on symmetry, and combine them via the join operation. Assuming that the dynamics of the component networks supports robust simple heteroclinic cycles or networks, we show that the join dynamics realizes a more complex heteroclinic network given by the product of those cycles or networks. Moreover, the equilibria in the product heteroclinic network correspond to partially synchronous states. Assuming no symmetry for the component coupled cell networks, one of the key points for the existence and robustness of the heteroclinic dynamics is the flow-invariant subspaces forced by the network structure – the synchrony subspaces. The other key point is that the (linear) stability of equilibria in the join dynamics is determined by the (linear) stability of equilibria in the component dynamics. The first point depends only at the network structure of the component networks. The second one depends both at the components network structures and the convenient choice of the join coupling. The proposed method is general and can be applied to the join of symmetric or asymmetric networks. Here, we illustrate it through the join of two asymmetric coupled cell networks where robust simple heteroclinic cycles between fully synchronous equilibria occur. We obtain robust simple heteroclinic networks for the join dynamics between partially synchronized equilibria for the associated join network.

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(ϕ) (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.

On designing heteroclinic networks from graphs

Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a complicated structure 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 directed graphs can be realised as attracting robust heteroclinic networks between states in phase space. This paper presents two methods of realising arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs—we say the ODEs realise the graphs as heteroclinic networks between equilibria that represent the vertices. Suppose we have a directed graph on nv vertices with ne edges. The “simplex realisation” embeds the graph as an invariant set of a flow on an (nv−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 (ne+1)-dimensional space. This method realises the graph as long as it is one-cycle free. In both cases we realise 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.

A heteroclinic network in mode interaction with symmetry

We study a robust heteroclinic network existing in generic mode interactions of symmetric dynamical systems. Each mode lies in C 3 and is equivariant under the action of D 6 ⋉ T 2 × Z 2. With this symmetry there are eight different types of non-trivial steady states. This work is motivated by Boussinesq convection on a plane layer with periodic boundary conditions on a hexagonal lattice. The mode interaction takes place in a centre eigenspace isomorphic to C 6 when the trivial steady state becomes unstable to two modes of the form of rolls with spatial periods in the ratio. Due to relations between the normal form coefficients, only four types of steady states can be involved in the network. We examine the normal form restricted to R 6, a flow-invariant subspace, then we describe the dynamics near the network and discuss subnetworks and switching near them.

A mechanism for switching near a heteroclinic network

We describe an example of a robust heteroclinic network for which nearby orbits exhibit irregular but sustained switching between the various sub-cycles in the network. The mechanism for switching is the presence of spiralling due to complex eigenvalues in the flow linearized about one of the equilibria common to all cycles in the network. We construct and use return maps to investigate the asymptotic stability of the network, and show that switching is ubiquitous near the network. Some of the unstable manifolds involved in the network are two-dimensional; we develop a technique to account for all trajectories on those manifolds. A simple numerical example illustrates the rich dynamics that can result from the interplay between the various cycles in the network.

Heteroclinic Ratchets in Networks of Coupled Oscillators

We analyze an example system of four coupled phase oscillators and discover a novel phenomenon that we call a “heteroclinic ratchet”; a particular type of robust heteroclinic network on a torus where connections wind in only one direction. The coupling structure has only one symmetry, but there are a number of invariant subspaces and degenerate bifurcations forced by the coupling structure, and we investigate these. We show that the system can have a robust attracting heteroclinic network that responds to a specific detuning Δ between certain pairs of oscillators by a breaking of phase locking for arbitrary Δ>0 but not for Δ≤0. Similarly, arbitrary small noise results in asymmetric desynchronization of certain pairs of oscillators, where particular oscillators have always larger frequency after the loss of synchronization. We call this heteroclinic network a heteroclinic ratchet because of its resemblance to a mechanical ratchet in terms of its dynamical consequences. We show that the existence of heteroclinic ratchets does not depend on symmetry or number of oscillators but depends on the specific connection structure of the coupled system.

Essentially asymptotically stable homoclinic networks

Melbourne [An example of a nonasymptotically stable attractor, Nonlinearity 4(3) (1991), pp. 835–844] discusses an example of a robust heteroclinic network that is not asymptotically stable but which has the strong attracting property called essential asymptotic stability. We establish that this phenomenon is possible for homoclinic networks, where all heteroclinic trajectories are symmetry related. Moreover, we study a transverse bifurcation from an asymptotically stable to an essentially asymptotically stable homoclinic network. The essentially asymptotically stable homoclinic network turns out to attract all nearby points except those on codimension-one stable manifolds of equilibria outside the homoclinic network.

The [formula omitted] mode interaction and heteroclinic networks in Boussinesq convection

Methods of equivariant bifurcation theory are applied to Boussinesq convection in a plane layer with stress-free horizontal boundaries and an imposed square lattice periodicity in the horizontal directions. We consider the problem near the onset of instability of the uniform conducting state where spatial roll patterns with two different wavelengths in the ratio 1 : 2 become simultaneously unstable at a mode interaction. Center manifold reduction yields a normal form on C 4 with very rich dynamical behavior. For a fixed Prandtl number P the mode interaction occurs at an isolated point in the parameter plane ( R , L ) (where R is the Rayleigh number and L the length of the horizontal periodicities) and acts as an organizing center for many nearby bifurcations. The normal form predicts the appearance of many steady states and travelling waves, which are classified by their symmetries. It also predicts the appearance of robust heteroclinic networks involving steady states with several different symmetries, and robust attractors of generalized heteroclinic type that include connections from equilibria to subcycles. This is the first example of a heteroclinic network in a fluid dynamical system that has ‘depth’ greater than one. The normal form dynamics is in good correspondence (both quantitatively and qualitatively) with direct numerical simulations of the full convection equations.

Regular and irregular cycling near a heteroclinic network

Heteroclinic networks are invariant sets containing more than one heteroclinic cycle. Such networks can appear robustly in equivariant vector fields. Previous authors have demonstrated that trajectories near heteroclinic networks can be attracted to one of a number of simultaneously ‘stable’ invariant subsets of the network. None of these invariant sets are asymptotically stable, but satisfy weaker definitions of stability. In this paper we discuss the behaviour of trajectories for one specific symmetric vector field. This vector field contains a robust heteroclinic network, and nearby trajectories display a variety of interesting dynamics. In particular, trajectories are observed to settle into a pattern of excursions around different parts of the network that we call ‘cycling sub-cycles’. Cycling patterns displaying different numbers of loops around the individual component cycles can be stable for the same parameter values, as can combinations of regular and irregular cycling. Analytic results for the regular cycling behaviour agree well with numerical simulations. We show that there exist parameter values where some trajectories display irregular cycling behaviour, in the sense that the numbers of loops around individual sub-cycles form a bounded aperiodic infinite sequence.