Existence Of Heteroclinic Cycles Research Articles

Heteroclinic Dynamics of Localized Frequency Synchrony: Heteroclinic Cycles for Small Populations

Many real-world systems can be modeled as networks of interacting oscillatory units. Collective dynamics that are of functional relevance for the oscillator network, such as switching between metastable states, arise through the interplay of network structure and interaction. Here, we give results for small networks on the existence of heteroclinic cycles between dynamically invariant sets on which the oscillators show localized frequency synchrony. Trajectories near these heteroclinic cycles will exhibit sequential switching of localized frequency synchrony: a population oscillators in the network will oscillate faster (or slower) than others and which population has this property sequentially changes over time. Since we give explicit conditions on the system parameters for such dynamics to arise, our results give insights into how network structure and interactions (which include higher-order interactions between oscillators) facilitate heteroclinic switching between localized frequency synchrony.

Sequential switching activity in ensembles of inhibitory coupled oscillators

The heteroclinic cycles and channels are mathematical images of a sequential switching activity in neural ensembles. In this paper we present a phenomenological model of such activity. The model is based on coupled Poincaré systems. The existence of heteroclinic cycles and channels is shown.

Robust heteroclinic cycles in delay differential equations

We consider the existence of heteroclinic cycles in Γ-equivariant delay-differential equations which emerge from symmetry-breaking bifurcations from an equilibrium solution with maximal isotropy subgroup. We begin by describing the existence of robust heteroclinic cycles on finite-dimensional centre manifolds and show that these are also robust to Γ-equivariant perturbations of the delay-differential equation. We then present the first example of a delay-differential equation which supports a heteroclinic cycle not contained within a finite-dimensional submanifold. This system is a delayed version of the Guckenheimer and Holmes equivariant three-dimensional example realized as a coupled cell system. We prove the existence of the heteroclinic cycle and show that it is structurally stable to Γ-equivariant perturbations which preserve certain codimension one subspaces of phase space associated with fixed point subspaces. By letting the cell dynamics be delay-dependent, we show that for a large enough delay, we obtain a heteroclinic cycle joining periodic solutions. Numerical simulations are presented and discussed.

Oscillations toward the Singularity of Locally Rotationally Symmetric Bianchi Type IX Cosmological Models with Vlasov Matter

We analyze the dynamics of a class of cosmological solutions of the Einstein–Vlasov equations. These equations describe an ensemble of collisionless particles (which represent galaxies or clusters of galaxies) that interact gravitatively through Einstein's equations of general relativity. The cosmological models we consider are spatially homogeneous, of Bianchi type IX, and locally rotationally symmetric (LRS). We prove that generic solutions exhibit an oscillatory behavior close to the singularities (the “big bang” in the past and the “big crunch” in the future); this is in contrast to the behavior of Einstein-vacuum or Einstein–Euler solutions. To establish this result we make use of dynamical systems theory; we introduce dimensionless dynamical variables that are defined on a compact state space. In this formulation the oscillatory behavior of generic solutions is represented by the existence of heteroclinic cycles on the boundary of the state space.

The GEOFLOW-experiment on ISS (Part III): Bifurcation analysis

Codimension 2 bifurcation of convective patterns in the nonrotating spherical Bénard Problem of Boussinesq fluid is studied using center manifold reduction. The aim is to determine, in the GEOFLOW-experiment framework, the physical parameters in order to obtain intermittent-like behaviour dynamics. We compute the critical aspect ratio and Rayleigh number corresponding to the (ℓ, ℓ+1) mode interaction. In the case of the (2,3) interaction, the existence of heteroclinic cycles are examined versus the Prandtl number.

Heteroclinic cycles and wreath product symmetries

We consider the existence and stability of heteroclinic cycles arising by local bifurcation in dynamical systems with wreath product symmetry = Z 2 G, where Z 2 acts by - 1 on R and G is a transitive subgroup of the permutation group S N (thus G has degree N). The group acts absolutely irreducibly on R N . We consider primary (codimension one) bifurcations from an equilibrium to heteroclinic cycles as real eigenvalues pass through zero. We relate the possibility of such cycles to the existence of non-gradient equivariant vector fields of cubic order. Using Hilbert series and the software package MAGMA we show that apart from the cyclic groups G (previously studied by other authors) only five groups G of degree h 7 are candidates for the existence of heteroclinic cycles. We establish the existence of certain types of heteroclinic cycle in these cases by making use of the concept of a subcycle. We also discusss edge cycles, and a generalization of heteroclinic cycles which we call a heteroclinic web. We apply our method to three examples.

Cyclical behavior in early universe cosmologies

This paper studies early universe cosmologies derived from a scalar–tensor action containing cosmological constant terms and massless fields. The governing equations can be written as a dynamical system which contains no past or future asymptotic equilibrium states (i.e., no sources nor sinks). This leads to dynamics with very interesting mathematical behavior such as the existence of heteroclinic cycles. The corresponding cosmologies have novel characteristics, including cyclical and bouncing behavior possibly indicating chaos. The connection between these early universe cosmologies and those derived from the low-energy string effective action is discussed.

The heteroclinic cycle in the model of competition betweenn species and its stability

This paper gives a sufficient condition for the existence of heteroclinic cycle in the model of competition betweenn species and a criterion for determining the stability of the heteroclinic cycle. The results given in this paper extend the results obtained by May and Leonard in and by Hofbauer and Sigmund in. A conjecture on the permanence of the model and a open problem on the stability of the heteroclinic cycle for the critical case are given at the end of this paper.

A systematic study of heteroclinic cycles in dynamical systems with broken symmetries

Let X be a Banach space or a manifold and G a compact Lie group acting on X. We study G-equivariant (semi)flows on X in the context of forced symmetry breaking. After applying small symmetry breaking perturbations, certain generic invariant manifolds of the above flows persist slightly changed. We obtain necessary and sufficient conditions for the existence of heteroclinic cycles on the perturbed manifolds. Applications are given for the case G = SO(3).