Regular and Chaotic Dynamics最新文献

We study the local dynamics of chains of coupled nonlinear systems of second-order ordinary differential equations of diffusion-difference type. The main assumption is that the number of elements of chains is large enough. This condition allows us to pass to the problem with a continuous spatial variable.Critical cases have been considered while studying the stability of the equilibrum state.It is shown that all these cases have infinite dimension. The research technique is based on the development and application of special methods for construction of normal forms.Among the main results of the paper, we include the creation of new nonlinear boundary value problems of parabolic type, whose nonlocal dynamics describes the local behavior of solutions of the original system.

This paper studies quasi-periodicity phenomena appearing at the transition from spiking to bursting activities in the Pernarowski model of pancreatic beta cells. Continuing the parameter, we show that the torus bifurcation is responsible for the transition between spiking and bursting. Our investigation involves different torus bifurcations, such as supercritical torus bifurcation, saddle torus canard, resonant torus, self-similar torus fractals, and torus destruction. These bifurcations give rise to complex or multistable dynamics. Despite being a dissipative system, the model still exhibits KAM tori, as we have illustrated. We provide two scenarios for the onset of resonant tori using the Poincaré return map, where global bifurcations happen because of the saddle-node or inverse period-doubling bifurcations. The blue-sky catastrophe takes place at the transition route from bursting to spiking.

We study nonconservative quasi-periodic (with (m) frequencies) perturbations of two-dimensional Hamiltonian systems with nonmonotonic rotation. It is assumed that the perturbation contains the so-called parametric terms. The behavior of solutions in the vicinity of degenerate resonances is described. Conditions for the existence of resonance ((m+1))-dimensional invariant tori for which there are no generating ones in the unperturbed system are found. The class of perturbations for which such tori can exist is indicated. The results are applied to the asymmetric Duffing equation under a parametric quasi-periodic perturbation.

Discharges of different epilepsies are characterized by different signal shape and duration.The authors adhere to the hypothesis that spike-wave discharges are long transient processes rather than attractors. This helps to explain some experimentally observed properties of discharges, including theabsence of a special termination mechanism and quasi-regularity.Analytical approaches mostly cannot be applied to studying transient dynamics in large networks. Therefore, to test the observed phenomena for universality one has to show that the same results can be achieved using different model types for nodes and different connectivity terms. Here, we study a class of simple networkmodels of a thalamocortical system and show that for the same connectivity matrices long, but finite in time quasi-regular processes mimicking epileptic spike-wave discharges can be found using nodes described by three neuron models: FitzHugh – Nagumo, Morris – Lecar and Hodgkin – Huxley. This resulttakes place both for linear and nonlinear sigmoid coupling.

Let (mathbb{G}_{k}^{cod1}(M^{n})), (kgeqslant 1), be the set of axiom A diffeomorphisms such thatthe nonwandering set of any (finmathbb{G}_{k}^{cod1}(M^{n})) consists of (k) orientable connected codimension one expanding attractors and contracting repellers where (M^{n}) is a closed orientable (n)-manifold, (ngeqslant 3). We classify the diffeomorphisms from (mathbb{G}_{k}^{cod1}(M^{n})) up to the global conjugacy on nonwandering sets. In addition, we show that any (finmathbb{G}_{k}^{cod1}(M^{n})) is (Omega)-stable and is not structurally stable. One describes the topological structure of a supporting manifold (M^{n}).

The focus of the work is the investigation of chaos and closely related dynamic properties of continuous actions of almost opensemigroups and (C)-semigroups. The class of dynamical systems ((S,X)) defined by such semigroups (S) is denoted by (mathfrak{A}).These semigroups contain, in particular, cascades, semiflows and groups of homeomorphisms. We extend the Devaney definition of chaos to general dynamical systems. For ((S,X)inmathfrak{A}) on locally compact metric spaces (X) with a countable base weprove that topological transitivity and density of the set formed by points having closed orbits imply the sensitivity to initial conditions. We assume neither the compactness of metric space nor the compactness of the above-mentioned closed orbits.In the case when the set of points having compact orbits is dense, our proof proceeds without the assumption of local compactness of the phase space (X). This statement generalizes the well-known result of J. Banks et al. on Devaney’s definitionof chaos for cascades.The interrelation of sensitivity, transitivity and the property of minimal sets of semigroups is investigated. Various examples are given.

We study a slow-fast system with two slow and one fast variables.We assume that the slow manifold of the system possesses a fold and there is an equilibrium of the system in a small neighborhood of the fold. We derive a normal form for the systemin a neighborhood of the pair “equilibrium-fold”and study the dynamics of the normal form. In particular, as the ratio of two time scales tends to zero we obtain an asymptotic formula for the Poincaré mapand calculate the parameter values for the first period-doubling bifurcation. The theory is applied to a generalization of the FitzHugh – Nagumo system.

We consider a topologically mixing hyperbolic attractor (Lambdasubset M^{n}) for a diffeomorphism (f:M^{n}to M^{n}) of a compact orientable (n)-manifold (M^{n}), (n>3). Such an attractor (Lambda) is called an Anosov torus provided the restriction (f|_{Lambda}) is conjugate to Anosov algebraic automorphism of (k)-dimensional torus (mathbb{T}^{k}).We prove that (Lambda) is an Anosov torus for two cases:1) (dim{Lambda}=n-1), (dim{W^{u}_{x}}=1), (xinLambda);2) (dimLambda=k,dim W^{u}_{x}=k-1,xinLambda), and (Lambda) belongs to an (f)-invariant closed (k)-manifold, (2leqslant kleqslant n), topologically embedded in (M^{n}).

The dynamics of two coupled neuron models, the Hindmarsh – Rose systems, are studied. Theirinteraction is simulated via a chemical coupling that is implemented with a sigmoid function.It is shown that the model may exhibit complex behavior: quasi-periodic, chaotic andhyperchaotic oscillations. A phenomenological scenario for the formation of hyperchaosassociated with the appearance of a discrete Shilnikov attractor is described. It is shownthat the formation of these attractors leads to the appearance of in-phase burstingoscillations.

We review a recent application of the ideas of normal form theory to systems (Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate variable. The main differencefrom the standard case consists in the non-uniqueness of the normal form and the total absence of the smalldivisors problem. The exposition is quite general, so as to allow extensions to the caseof more non-periodic coordinates, and more functional settings. Here, for simplicity,we work in the real-analytic class.