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.     

Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation

We 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.     

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.     

6-Periodic travelling waves in an artificial neural network with bang–bang control

Doubly periodic travelling waves can be used to describe dynamic patterns of signals that govern movements of animals. In this paper, we study the existence of such waves in cellular networks involving the discontinuous Heaviside step function. This is done by finding ω-periodic solutions of an accompanying recurrence relation with a priori unknown parameters and the Heaviside function. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by means of symmetry, combinatorial techniques and accompanying linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 6. Our techniques are new and good for other periodic solutions with relatively small periods.     

The Theory of Linear G-Difference Equations

We introduce the notion of difference equations defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations, symmetries of an equation are module endomorphisms, and conserved structures are invariants in the tensor algebra of the given equation.We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions include these as a special case.     

Target Patterns and Horseshoes from a Perturbed Central-Force Problem: Some Temporally Periodic Solutions to Reaction-Diffusion Equations

Existence and construction of some time-periodic solutions of a class of reaction-diffusion equations is described. These give examples of center-structures from which emanate trains of waves with outward directed group velocity, in an infinite one-dimensional spatial domain. A discrete set of distinct types of center-structures is found. Similar results are found for sufficiently large finite regions with impermeable boundaries. The existence of many other time periodic solutions corresponding to spatially infinite irregular arrays of center-structures of different types is also demonstrated for these systems. Some numerical examples are presented.     

The Geometry of the Phase Diffusion Equation

$ \tau(|{{\vec k}}|) \mbox{\bf $\Theta$}_T = -\nabla\cdot (B(|{{\vec k}}|)\cdot {{\vec k}}), \,\, {{\vec k}} = \nabla \mbox{\bf $\Theta$},$ and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order ``self-dual' equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. Received on October 30, 1998; final revision received July 6, 1999     

The dynamics ofn weakly coupled identical oscillators

Summary We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.     

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.     

Symmetries of periodic solutions for planar potential systems

In this article we discuss the symmetries of periodic solutions to Hamiltonian systems with two degrees of freedom in mechanical form. The possible symmetries of such periodic trajectories are generated by spatial symmetries (a finite subgroup of , phase-shift symmetries (the circle group , and a time-reversing symmetry (associated with mechanical form). We focus on the symmetries and structures of the trajectories in configuration space (), showing that special properties such as self-intersections and brake orbits are consequences of symmetry.

     

Exact solutions and dynamics of the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity

This paper investigate the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity by using the method of dynamical systems. The functions $q(x,t)=\phi(\xi)\exp(i(-kx+\omega t))$ are solutions of the equation (1.1) that governs the propagation of Raman solitons through optical metamaterials, where $\xi=x-vt$ and $\phi(\xi)$ in the solutions satisfy a singular planar dynamical system (1.5) which has two singular straight lines. By using the bifurcation theory method of dynamical systems to the equation of $\phi(\xi)$, bifurcations of phase portraits for this dynamical system are obtained under 28 different parameter conditions. Based on those phase portraits, 62 exact solutions of system (1.5) including periodic solutions, heteroclinic and homoclinic solutions, periodic peakons and peakons as well as compacton solutions are derived.     

Some remarks on period doubling in systems with symmetry

Period doubling of periodic solutions in systems with symmetry leads to certain group theoretical difficulties, if a periodic solution possesses a mixed spatio-temporal symmetry. Based on a result of Vanderbauwhede [11] on period doubling with symmetry a method is presented to determine systematically the bifurcations that one may expect in such a system. The results are used to analyse multiple period doublings of periodic solutions with dihedral group symmetry.     

Coupled nonlinear oscillators and the symmetries of animal gaits

Summary Animal locomotion typically employs several distinct periodic patterns of leg movements, known as gaits. It has long been observed that most gaits possess a degree of symmetry. Our aim is to draw attention to some remarkable parallels between the generalities of coupled nonlinear oscillators and the observed symmetries of gaits, and to describe how this observation might impose constraints on the general structure of the neural circuits, i.e. central pattern generators, that control locomotion. We compare the symmetries of gaits with the symmetry-breaking oscillation patterns that should be expected in various networks of symmetrically coupled nonlinear oscillators. We discuss the possibility that transitions between gaits may be modeled as symmetry-breaking bifurcations of such oscillator networks. The emphasis is on general model-independent features of such networks, rather than on specific models. Each type of network generates a characteristic set of gait symmetries, so our results may be interpreted as an analysis of the general structure required of a central pattern generator in order to produce the types of gait observed in the natural world. The approach leads to natural hierarchies of gaits, ordered by symmetry, and to natural sequences of gait bifurcations. We briefly discuss how the ideas could be extended to hexapodal gaits.     

Application of mode decomposition approach to the synchronization of the non-identical dynamical systems

The synchronization problem of two different dynamical systems is considered by employing mode decomposition approach in this paper. Synchronization of non-identical coupled dynamical systems with non-chaotic attractors, i.e., equilibria, periodic and quasi-periodic solutions, is investigated analytically and numerically. Some results are obtained by this method. Some examples, supported by numerical simulation, are presented to illustrate the conciseness and effectiveness of the approach.     

Explicit periodic travelling waves for a discrete lambda–omega reaction–diffusion system

In 1973, Kopell and Howard introduced a λ–ω reaction–diffusion system and found an explicit family of periodic travelling wave solutions lying on circles with radius less than 1. Since λ–ω systems represent universal models for studying chemical processes, and onset of turbulent behaviour, etc., explicit solutions of λ–ω systems with delays or discrete λ–ω systems can be of further help when the only method for obtaining other solutions is through numerical computation. There are now much investigations of various λ–ω systems. However, it is of interest to note that none attempts to find explicit travelling wave solutions. In this paper, we investigate the existence of explicit solutions for the simplest Euler scheme of a λ–ω system with delays or advancements which is described as a coupled pair of partial difference equations. We are able to provide necessary as well as sufficient conditions for the existence of numerical periodic travelling wave solutions. Additionally, we also provide some examples to show that our explicit solutions are qualitatively different from those found by Kopell and Howard and hence they may be of interests for specialists in the area of reaction–diffusion systems.     

Affine-Periodic Solutions and Pseudo Affine-Periodic Solutions for Differential Equations with Exponential Dichotomy and Exponential Trichotomy

It is proved that every $(Q,T)$-affine-periodic differential equation has a $(Q,T)$-affine-periodic solution if the corresponding homogeneous linear equation admits exponential dichotomy or exponential trichotomy. This kind of ``periodic'' solutions might be usual periodic or quasi-periodic ones if $Q$ is an identity matrix or orthogonal matrix. Hence solutions also possess certain symmetry in geometry. The result is also extended to the case of pseudo affine-periodic solutions.     

Hopf bifurcation and multiple periodic solutions in Lotka–Volterra systems with symmetries

The purpose of this paper is to study Hopf bifurcations in a delayed Lotka–Volterra system with dihedral symmetry. By treating the response delay as bifurcation parameter and employing equivariant degree method, we obtain the existence of multiple branches of nonconstant periodic solutions through a local Hopf bifurcation around an equilibrium. We find that competing coefficients and the response delay in the system can affect the spatio-temporal patterns of bifurcating periodic solutions. According to their symmetric properties, a topological classification is given for these periodic solutions. Furthermore, an estimation is presented on minimal number of bifurcating branches. These theoretical results are helpful to better understand the complex dynamics induced by response delays and symmetries in Lotka–Volterra systems.     

Bifurcations and exact travelling wave solutions of M-N-Wang equation

By using the method of dynamical systems to Mikhailov-Novikov-Wang Equation, through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system of the derivative $\phi(\xi)$ of the wave function $\psi(\xi)$. Under different parameter conditions, for $\phi(\xi)$, exact explicit solitary wave solutions, periodic peakon and anti-peakon solutions are obtained. By integrating known $\phi(\xi)$, nine exact explicit traveling wave solutions of $\psi(\xi)$ are given.     

Recurrence and symmetry of time series: Application to transition detection

The study of transitions in low dimensional, nonlinear dynamical systems is a complex problem for which there is not yet a simple, global numerical method able to detect chaos–chaos, chaos–periodic bifurcations and symmetry-breaking, symmetry-increasing bifurcations. We present here for the first time a general framework focusing on the symmetry concept of time series that at the same time reveals new kinds of recurrence. We propose several numerical tools based on the symmetry concept allowing both the qualification and quantification of different kinds of possible symmetry. By using several examples based on periodic symmetrical time series and on logistic and cubic maps, we show that it is possible with simple numerical tools to detect a large number of bifurcations of chaos–chaos, chaos–periodic, broken symmetry and increased symmetry types.