Heteroclinic Cycles in Hopfield Networks

It is widely believed that information is stored in the brain by means of the varying strength of synaptic connections between neurons. Stored patterns can be replayed upon the arrival of an appropriate stimulus. Hence, it is interesting to understand how an information pattern can be represented by the dynamics of the system. In this work, we consider a class of network neuron models, known as Hopfield networks, with a learning rule which consists of transforming an information string to a coupling pattern. Within this class of models, we study dynamic patterns, known as robust heteroclinic cycles, and establish a tight connection between their existence and the structure of the coupling.     

Heteroclinic Cycles in Coupled Systems of Difference Equations

Cycling behavior, in which solution trajectories linger around steady-states and periodic solutions, is known to be a generic feature of coupled cell systems of differential equations. In this type of systems, cycling behavior can even occur independently of the internal dynamics of each cell. This conclusion has lead to the discovery of "cycling chaos", in which solution trajectories cycle around symmetrically related chaotic sets. In this work, we demonstrate that cycling behavior also occurs in coupled systems of difference equations. More specifically, we prove the existence of structurally stable cycles between fixed points, and use numerical simulations to illustrate that the resulting cycles can also persist independently of the internal dynamics of each cell. Consequently, we demonstrate that cycles involving periodic orbits as well as cycling chaos also occur in systems of difference equations.     

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.     

Feedforward Networks: Adaptation,Feedback, and Synchrony

Journal of Nonlinear Science - We investigate the effect on synchrony of adding feedback loops and adaptation to a large class of feedforward networks. We obtain relatively complete results on...     

Designing Heteroclinic and Excitable Networks in Phase Space Using Two Populations of Coupled Cells

We give a constructive method for realising 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.     

The Lattice of Synchrony Subspaces of a Coupled Cell Network: Characterization and Computation Algorithm

Coupled cell systems are networks of dynamical systems (the cells), where the links between the cells are described through the network structure, the coupled cell network. Synchrony subspaces are spaces defined in terms of equalities of certain cell coordinates that are flow-invariant for all coupled cell systems associated with a given network structure. The intersection of synchrony subspaces of a network is also a synchrony subspace of the network. It follows, then, that, given a coupled cell network, its set of synchrony subspaces, taking the inclusion partial order relation, forms a lattice. In this paper we show how to obtain the lattice of synchrony subspaces for a general network and present an algorithm that generates that lattice. We prove that this problem is reduced to obtain the lattice of synchrony subspaces for regular networks. For a regular network we obtain the lattice of synchrony subspaces based on the eigenvalue structure of the network adjacency matrix.     

Almost Complete and Equable Heteroclinic Networks

Journal of Nonlinear Science - Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria,...     

Heteroclinic and Periodic Cycles in a Perturbed Convection Model

Vivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast - slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) - solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov-Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity.     

Some Curious Phenomena in Coupled Cell Networks

We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a three-cell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic 2n × 2n arrays of cells that generate 2ⁿ different patterns of synchrony from one symmetry-generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form.     

Generalized Heteroclinic Cycles in Spherically Invariant Systems and Their Perturbations

Summary. In this paper we want to investigate the effects of forced symmetry-breaking perturbations—see Lauterbach & Roberts [29], as well as [28], [31]—on the heteroclinic cycle which was found in the l = 1 , l = 2 mode interaction by Armbruster and Chossat [1], [12] and generalized by Chossat and Guyard [25], [14]. We show that this cycle is embedded in a larger class of cycles, which we call a generalized heteroclinic cycle (GHC). After describing the structure of this set, we discuss its stability. Then the persistence under symmetry-breaking perturbations is investigated. We will discuss also the application to the spherical Bénard problem, which was the initial motivation for this work. Received March 11, 1997; first revision received October 10, 1997; second revision received April 13, 1998; accepted July 16, 1998     

Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc $\mathcal {D}$ whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in $\mathcal {D}$ , which is invariant under the action of a lattice subgroup ?? of U(1,1), the group of isometries of ${\mathcal{D}}$ . In our case ?? generates a tiling of $\mathcal {D}$ with regular octagons. This problem was introduced as an example of spontaneous pattern formation in a model of image feature detection by the visual cortex where the features are assumed to be represented in the space of structure tensors. Under ??generic?? assumptions the bifurcation problem reduces to an ODE which is invariant by an irreducible representation of the group of automorphisms $\mathcal {G}$ of the compact Riemann surface $\mathcal {D}/\varGamma $ . The irreducible representations of $\mathcal {G}$ have dimensions one, two, three and four. The bifurcation diagrams for the representations of dimensions less than four have already been described and correspond to well-known group actions. In the present work we compute the bifurcation diagrams for the remaining three irreducible representations of dimension four, thus completing the classification. In one of these cases, there is generic bifurcation of a heteroclinic network connecting equilibria with two different orbit types.     

一类异宿环分支极限环的唯一性

本文研究平面上一类两点或三点异宿环附近极限环的分支,在一简洁条件下证明了异宿环分支极限环的唯一性,并给出了极限环唯一存在的充要条件.作为对三维余维2分支的应用,解决了所出现的两点异宿环产生唯一极限环的问题.     

Banach空间中正则非退化异宿环的Lipschitz 扰动

本文研究Banach空间上离散动力系统的Lipschitz扰动.设f,g是Banach空间(X,||·||)上的连续自映射.如果f具有正则非退化返回排斥子或正则非退化异宿环且g是f的Lipschitz小扰动,则g也有正则非退化返回排斥子或正则非退化异宿环.另外,本文还证明完备度量空间中正则非退化异宿环蕴含正则非退化返回...     

Symmetry Groupoids and Admissible Vector Fields for Coupled Cell Networks

The space of admissible vector fields, consistent with the structureof a network of coupled dynamical systems, can be specifiedin terms of the network's symmetry groupoid. The symmetry groupoidalso determines the robust patterns of synchrony in the network– those that arise because of the network topology. Inparticular, synchronous cells can be identified in a canonicalmanner to yield a quotient network. Admissible vector fieldson the original network induce admissible vector fields on thequotient, and any dynamical state of such an induced vectorfield can be lifted to the original network, yielding an analogousstate in which certain sets of cells are synchronized. In thepaper, necessary and sufficient conditions are specified forall admissible vector fields on the quotient to lift in thismanner. These conditions are combinatorial in nature, and theproof uses invariant theory for the symmetric group. Also thesymmetry groupoid of a quotient is related to that of the originalnetwork, and it is shown that there is a close analogy withthe usual normalizer symmetry that arises in group-equivariantdynamics.     

Heteroclinic Solutions for a System of Strongly Coupled ODEs

We use the implicit function theorem to prove an existence of a heteroclinic orbit to a system of two non-linear second-order ODEs. The perturbation is carried out around infinite value of a ‘coupling parameter’. The form of the system which is considered in this paper is related to the system defining travelling wave solutions in a two temperature model of the laser sustained plasma.     

Synchronization of Coupled Limit Cycles

A unified approach to the analysis of synchronization in coupled systems of autonomous differential equations is presented in this work. Through a careful analysis of the variational equation of the coupled system we establish a sufficient condition for synchronization in terms of the geometric properties of the local limit cycles and the coupling operator. This result applies to a large class of differential equation models in physics and biology. The stability analysis is complemented by a discussion of numerical simulations of a compartmental model of a neuron.     

Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in $${\mathbb R}^4$$

Robust heteroclinic cycles in equivariant dynamical systems in \({\mathbb R}^4\) have been a subject of intense scientific investigation because, unlike heteroclinic cycles in \({\mathbb R}^3\), they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have compiled an exhaustive list of finite subgroups of O(4) admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilibrium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyze the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in \({\mathbb R}^4\).     

Existence of Generalized Heteroclinic Solutions of the Coupled Schr¨odinger System under a Small Perturbation

The following coupled Schrodinger system with a small perturbation
is considered, where β and ε are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution （called the generalized heteroclinic solution thereafter）.