Regular and Chaotic Dynamics最新文献

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.

The (2)-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the (2)-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the (2)-body problem on hyperbolic 3-space and discuss their stability for a strictly attractive potential.

A general method is presented for constructing a nonlinear canonical transformation, which makes it possible to introduce local variables in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. This method can be used for investigating the behavior of the Hamiltonian system inthe vicinity of its periodic trajectories. In particular, it can be applied to solve the problem of orbital stability of periodic motions.

In this paper, we obtain a classification of gradient-likeflows on arbitrary surfaces by generalizing the circularFleitasscheme. In 1975 he proved that such a scheme is a completeinvariant of topological equivalence for polar flows on 2- and 3-manifolds.In this paper, we generalize the concept of a circular schemeto arbitrary gradient-like flows on surfaces. We prove that theisomorphism class of such schemes is a complete invariant oftopological equivalence. We also solve exhaustively therealization problem by describing an abstract circularscheme and the process of realizing a gradient-like flow onthe surface. In addition, we construct an efficient algorithmfor distinguishing the isomorphism of circular schemes.

This paper treats the problem of a spherical robot with an axisymmetric pendulum driverolling without slipping on a vibrating plane. The main purpose of the paper isto investigate the stabilization of the upper vertical rotations of the pendulumusing feedback (additional control action). For the chosen type of feedback,regions of asymptotic stability of the upper vertical rotations of the pendulum are constructedand possible bifurcations are analyzed. Special attention is also given to the question ofthe stability of periodic solutions arising as the vertical rotations lose stability.

The space of (2)-jets of a real function of two real variables, denoted by (J^{2}(mathbb{R}^{2},mathbb{R})), admits the structure of a metabelian Carnot group, so (J^{2}(mathbb{R}^{2},mathbb{R})) has a normal abelian sub-group (mathbb{A}). As any sub-Riemannian manifold, (J^{2}(mathbb{R}^{2},mathbb{R})) has an associated Hamiltonian geodesic flow. The Hamiltonian action of (mathbb{A}) on (T^{*}J^{2}(mathbb{R}^{2},mathbb{R})) yields the reduced Hamiltonian (H_{mu}) on (T^{*}mathcal{H}simeq T^{*}(J^{2}(mathbb{R}^{2},mathbb{R})/mathbb{A})), where (H_{mu}) is a two-dimensional Euclidean space. The paper is devoted to proving that the reduced Hamiltonian (H_{mu}) is non-integrable by meromorphic functions for some values of (mu). This result suggests the sub-Riemannian geodesic flow on (J^{2}(mathbb{R}^{2},mathbb{R})) is not meromorphically integrable.