Regular and Chaotic Dynamics最新文献

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.

We consider a one-parameter family (f_{mu}) of multidimensional diffeomorphisms such that for (mu=0) the diffeomorphism (f_{0}) has a transversal homoclinic orbit to a nonhyperbolic fixed point of arbitrary finite order (ngeqslant 1) of degeneracy, and for (mu>0) the fixed point becomes a hyperbolic saddle. In the paper, we give a complete description of the structure of the set (N_{mu}) of all orbits entirely lying in a sufficiently small fixed neighborhood of the homoclinic orbit. Moreover, we show that for (mugeqslant 0) the set (N_{mu}) is hyperbolic (for (mu=0) it is nonuniformly hyperbolic) and the dynamical system (f_{mu}bigl{|}_{N_{mu}}) (the restriction of (f_{mu}) to (N_{mu})) is topologically conjugate to a certain nontrivial subsystem of the topological Bernoulli scheme of two symbols.

The purpose of this work is to study small oscillations and oscillations with an asymptotically large amplitude in nonlinear systems of two equations with delay, regularly depending on a small parameter. We assume that the nonlinearity is compactly supported, i. e., its action is carried out only in a certain finite region of phase space.Local oscillations are studied by classical methods of bifurcation theory, and the study of nonlocal dynamics is based on a special large-parameter method, which makes it possible to reduce the original problem to the analysis of a specially constructed finite-dimensional mapping. In all cases, algorithms for constructing the asymptotic behavior of solutions are developed. In the case of local analysis, normal forms are constructed that determine the dynamics of the original system in a neighborhood of the zero equilibrium state, the asymptotic behavior of the periodic solution is constructed, and the question of its stability is answered. In studying nonlocalsolutions, one-dimensional mappings are constructed that make it possible to determinethe behavior of solutions with an asymptotically large amplitude. Conditions for theexistence of a periodic solution are found and its stability is investigated.

We study the dynamics of a multidimensional slow-fast Hamiltonian system in a neighborhoodof the slow manifold under the assumption that the frozen system has a hyperbolic equilibriumwith complex simple leading eigenvaluesand there exists a transverse homoclinic orbit.We obtain formulas for the corresponding Shilnikov separatrix mapand prove the existence of trajectories in a neighborhood of the homoclinic setwith a prescribed evolution of the slow variables.An application to the (3) body problem is given.