Regular and Chaotic Dynamics最新文献

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.

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.

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 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.

This paper describes an approach to constructing the asymptotics of Gaussian beams, based on the theory of the canonical Maslov operator and the study of the dynamics and singularities of the corresponding Lagrangian manifolds in the phase space. As an example, we construct global asymptotics of Laguerre – Gauss beams, which are solutions of the Helmholtz equation in the paraxial approximation. Depending on the type of the beam and the emerging singularity on the Lagrangian manifold, asymptotics are expressed in terms of the Airy function or the Bessel function. One of the advantages of the described approach is that we can abandon the paraxial approximation and construct global asymptotics in terms of special functions also for solutions of the original Helmholtz equation, which is illustrated by an example.

An arbitrary diffeomorphism (f) of class (C^{1}) acting from an open subset (U) of Riemannian manifold (M) of dimension (m,) (mgeqslant 2,) into (f(U)subset M) is considered.Let (A) be a compact subset of (U) invariant for (f,) i. e., (f(A)=A.)Various sufficient conditions are proposed under which (A) is a hyperbolic set of the diffeomorphism (f.)

We consider the effect of an external periodic force on chimera states in the phase oscillator model proposed in [Phys. Rev. Lett, v. 101, 00319007 (2008)].Using the Ott – Antonsen reduction, the dynamical equations for the global order parametercharacterizing the degree of synchronization are constructed. The frequency locking by an external periodic force region isconstructed. The possibility of stable chimeras synchronization and unstable chimerasstabilization is established. The instability development of the chimera states leads to the appearance of breather chimeras or complete synchronization.