Regular and Chaotic Dynamics最新文献

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.

This paper is concerned with the plane-parallelmotion of an elliptic foil with an attached vortex ofconstant strength in an ideal fluid.Special attention is given to the case in which the vortexlies on the continuation of one of the semiaxes of the ellipse. It is shownthat in this case there exist no attracting solutions andthe system is integrable by the Euler – Jacobi theorem.A complete qualitative analysis of the equations ofmotion is carried out for cases where the vortex lies on the continuation ofthe large or the small semiaxis of the ellipse.Possible types of trajectories of an elliptic foil with an attachedvortex are established: quasi-periodic, unbounded(going to infinity) and periodic trajectories.

This paper addresses the problem of a homogeneous ball rolling on the inner surface of acircular cylinder in a field of gravity parallel to its axis. It is assumed that the ballrolls without slipping on the surface of the cylinder, and that the cylinder executesplane-parallel motions in a circle perpendicular to its symmetry axis. The integrability ofthe problem by quadratures is proved. It is shown that in this problem the trajectories ofthe ball are quasi-periodic in the general case, and that an unbounded elevation of the ballis impossible. However, in contrast to a fixed (or rotating) cylinder, there exist resonancesat which the ball moves on average downward with constant acceleration.

A two-layer quasigeostrophic model is considered in the (f)-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case where the structure consists of a central vortex of arbitrary effective intensity (Gamma) and (N) ((N=4,5) and (6)) identical peripheral vortices. The identical vortices, each having a unit effective intensity, are uniformly distributed over a circle of radius (R) in the lower layer. The central vortex lies either in the same or in another layer. The problem has three parameters ((R,Gamma,alpha)), where (alpha) is the difference between layer nondimensional thicknesses. The cases (N=2,3) were investigated by us earlier.

The theory of stability of steady-state motions of dynamical systems with a continuous symmetry group (mathcal{G}) is applied. The two definitions of stability used in the study are Routh stability and (mathcal{G})-stability.The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of avortex structure, and the (mathcal{G})-stability is the stability of a three-parameter invariant set (O_{mathcal{G}}), formed by the orbits of a continuous family of steady-state rotations of a two-layer vortex structure.The problem of Routh stability is reduced to the problem of stability of a family ofequilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically.

The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.

We prove that an (n)-sphere (mathbb{S}^{n}), (ngeqslant 2), admits structurally stable diffeomorphisms (mathbb{S}^{n}tomathbb{S}^{n}) with nonorientable expanding attractors of any topological dimension (din{1,ldots,[frac{n}{2}]}) where ([x]) is the integer part of (x). In addition, any (n)-sphere (mathbb{S}^{n}), (ngeqslant 3), admits axiom A diffeomorphisms (mathbb{S}^{n}tomathbb{S}^{n}) with orientable expanding attractors of any topological dimension (din{1,ldots,[frac{n}{3}]}). We prove that an (n)-torus (mathbb{T}^{n}), (ngeqslant 2), admits structurally stable diffeomorphisms (mathbb{T}^{n}tomathbb{T}^{n}) with orientable expanding attractors of any topological dimension (din{1,ldots,n-1}). We also prove that, given any closed (n)-manifold (M^{n}), (ngeqslant 2), and any (din{1,ldots,[frac{n}{2}]}), there is an axiom A diffeomorphism (f:M^{n}to M^{n}) with a (d)-dimensional nonorientable expanding attractor. Similar statements hold for axiom A flows.

We study a class of integrable inhomogeneous Lotka – Volterra systems whose quadratic terms are defined by an antisymmetric matrix and whose linear terms consist of three blocks. We provide the Poisson algebra of their Darboux polynomials and prove a contraction theorem. We then use these results to classify the systems according to the number of functionally independent (and, for some, commuting) integrals. We also establish separability/solvability by quadratures, given the solutions to the 2- and 3-dimensional systems, which we provide in terms of the Lambert (W) function.

It is known that there is a duality between the Davey – Stewartson type coupled systems and a class of integrable two-dimensional Toda type lattices. More precisely, the coupled systems are generalized symmetries for the lattices and the lattices can be interpreted as dressing chains for the systems. In our recent study we have found a novel lattice which is apparently not related to the known ones by Miura type transformation. In this article we describe higher symmetries to this lattice and derive a new coupled system of DS type.

We discuss some families of integrable and superintegrable systems in (n)-dimensional Euclidean space which are invariant under (mgeqslant n-2) rotations. The invariant Hamiltonian (H=sum p_{i}^{2}+V(q)) is integrable with (n-2) integrals of motion (M_{alpha}) and an additional integral ofmotion (G), which are first- and fourth-order polynomials in momenta, respectively.