Regular and Chaotic Dynamics最新文献

We describe a (C^{1})-open set of systems of differential equations in (R^{n}), for any (ngeqslant 4), where every system has a chain-transitive chaotic attractor whichcontains a saddle-focus equilibrium with a two-dimensional unstable manifold. The attractor also includes a wild hyperbolic set and a heterodimensional cycle involvinghyperbolic sets with different numbers of positive Lyapunov exponents.

S. Smale has shown that any closed smooth manifold admits a gradient-like flow, which is a structurally stable flow with a finite nonwandering set. Polar flows form a subclass of gradient-like flows characterized by the simplest nonwandering set for the given manifold, consisting of exactly one source, one sink, and a finite number of saddle equilibria. We describe the topology of four-dimensional closed manifolds that admit polar flows without heteroclinic intersections, as well as all classes of topological equivalence of polar flows on each manifold. In particular, we demonstrate that there exists a countable set of nonequivalent flows with a given number (kgeqslant 2) of saddle equilibria on each manifold, which contrasts with the situation in lower-dimensional analogues.

In investigating dynamical systems with chaotic attractors, many aspects of global behavior of a flow or a diffeomorphism with such an attractor are studied by replacing a nontrivial attractor by a trivial one [1, 2, 11, 14].Such a method allows one to reduce the original system to a regular system, for instance, of a Morse – Smale system, matched with it. In most cases, the possibility of such a substitution is justified by the existence of Morse – Smale diffeomorphisms with partially determined periodic data, the complete understanding of their dynamics and the topology of manifolds, on which they are defined. With this aim in mind, we consider Morse – Smale diffeomorphisms (f) with determined periods of the sink points, given on a closed smooth 3-manifold. We have shown that, if the total number of these sinks is (k), then their nonwandering set consists of an even number of points which is at least (2k). We have found necessary and sufficient conditions for the realizability of a set of sink periods in the minimal nonwandering set. We claim that such diffeomorphisms exist only on the 3-sphere and establish for them a sufficient condition for the existence of heteroclinic points. In addition, we prove that the Morse – Smale 3-diffeomorphism with an arbitrary set of sink periods can be implemented in the nonwandering set consisting of (2k+2) points. We claim that any such a diffeomorphism is supported by a lens space or the skew product (mathbb{S}^{2}tilde{times}mathbb{S}^{1}).

We introduce a class of examples which provide an affine generalization of the nonholonomic problem of a convex body that rolls without slipping on the plane. These examples are constructed by taking as given two vector fields, one on the surface of the body and another on the plane, which specify the velocity of the contact point. We investigate dynamical aspects of the system such as existence of first integrals, smooth invariant measure, integrabilityand chaotic behavior, giving special attention to special shapes of the convex body and specific choices of the vector fields for which the affine nonholonomic constraints may be physically realized.

Kolmogorov – Arnold – Moser theory started in the 1950s as theperturbation theory for persistence of multi- orquasi-periodic motions in Hamiltonian systems.Since then the theory obtained a branch where the persistentoccurrence of quasi-periodicity is studied in variousclasses of systems, which may depend on parameters.The view changed into the direction of structural stability,concerning the occurrence of quasi-periodic tori on a setof positive Hausdorff measure in a sub-manifold of theproduct of phase space and parameter space.This paper contains an overview of this development withan emphasis on the world of dissipative systems, wherefamilies of quasi-periodic tori occur and bifurcate in apersistent way.The transition from orderly to chaotic dynamics here formsa leading thought.

Continuous actions of topological semigroups on products (X) of an arbitrary family of topological spaces (X_{i}), (iin J,) are studied. The relationship between the dynamical properties of semigroups acting on the factors (X_{i}) and the same properties of the product of semigroups on the product (X) of these spaces is investigated. We consider the following dynamical properties: topological transitivity, existence of a dense orbit, density of a union of minimal sets, and density of the set of points with closed orbits. The sensitive dependence on initial conditions is investigated for countable products of metric spaces. Various examples are constructed. In particular, on an infinite-dimensional torus we have constructed a continualfamily of chaotic semigroup dynamical systemsthat are pairwise topologically not conjugate by homeomorphisms preserving the structure of theproduct of this torus.

This paper deals with the problem of smoothness of the stable invariant foliation for a homoclinic bifurcation with a neutral saddle in symmetric systems of differential equations. We give animproved sufficient condition for the existence of an invariant smooth foliation on a cross-section transversal to the stable manifold of the saddle. It is shown that the smoothness of the invariant foliation depends on the gap between the leading stable eigenvalue of the saddle and other stable eigenvalues. We also obtain an equation to describe the one-dimensional factor map, and we study the renormalization properties of this map. The improved information on the smoothness of the foliation and the factor map allows one to extend Shilnikov’s results on the birth of Lorenz attractors under the bifurcation considered.

In this paper we prove criteria of a (C^{0})- (Omega)-blowup in (C^{1})-smooth skew products with aclosed set of periodic points on multidimensional cells and give examples of maps that admit such a (Omega)-blowup.Our method is based on the study of the properties of the set of chain-recurrent points. We alsoprove that the set of weakly nonwandering points of maps under consideration coincides withthe chain-recurrent set, investigate the approximation (in the (C^{0})-norm) and entropy propertiesof (C^{1})-smooth skew products with a closed set of periodic points.

This paper provides an overview of the results obtained from the study of adaptive dynamical networks of Kuramoto oscillators with higher-order interactions. The main focus is on results in the field of synchronization and collective chaotic dynamics. Identifying the dynamical mechanisms underlying the synchronization of oscillator ensembles with higher-order interactions may contribute to further advances in understanding the work of some complex systems such as the neural networks of the brain.