Regular and Chaotic Dynamics最新文献

For a (mathcal{PT})-symmetric periodic Schrödinger operator, which is a small perturbation of the zero potential, we calculate the spectrum and the divisor of zeroes of the Bloch function in the leading order of perturbation theory. In particular, we show that the analogs of lacunae of the Bloch spectrum are ellipses, and their focal points coincide with the branch points of the spectral curve.

A Jacobi field is a potential force field whose potential is a homogeneous function of degree(-2). The problem of the motion of a particle in such a field admits an additional integralquadratic in velocities. It can be used to reduce the number of degrees of freedom and topass to the study of a reduced system with spherical configuration space. These resultsare extended to the more general case of the motion of a particlein spaces of constant curvature. An analysis is made of particle motionon a cone whose vertex coincides with the singular point of theJacobi potential. A lower estimate of the distance from the moving particle to the vertex of the cone is given. This approach is also applicable to a more general case where the charged particleis additionally located in the magnetic field of a monopole. A billiard inside the conewith a particle bouncing elastically off its boundary isconsidered.

A system of two diffusively coupled Bautin (generalized Stuart – Landau) oscillators is considered. Using a specially designed reduced system, the existence and stability of homogeneous solutions are investigated. Such solutions represent oscillatory regimes in which the amplitudes of different oscillators are identical to each other and coincide at any given time. A partition of “coupling strength — frequency mismatch” parameter plane into regions with different dynamical behavior of the oscillators is obtained. It is established that the phase space of the system has a foliation into a continuum of two-dimensional invariant manifolds. It is shown that oscillation quenching in the system, in contrast to systems of diffusively coupled Stuart – Landau oscillators, is determined by new mechanisms and is associated with the bifurcation of merger of invariant tori and the saddle-node (tangent) bifurcations of limit cycles. At the same time, the quenching does not occur monotonously with a change in the coupling strength, but abruptly, and the critical value of the coupling strength depends on the frequency mismatch between the oscillators.

We consider the problem of spectral stability of traveling wave solutions (u=gamma(x-Wt))for a system of viscous conservation laws (partial_{t}u+partial_{x}F(u)=partial^{2}_{x}u).Such solutions correspond to heteroclinic trajectories (gamma) of a system of ODEs.In general conditions of stability can be obtained only numerically.We propose a model class of piecewise linear (discontinuous) vector fields (F) for which thestabilityproblem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and constructseveral examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.

A treatment is given of the spatial restricted elliptic problem of threebodies interacting under Newtonian gravity. The problem depends on two parameters:the ratio between the masses of the main attracting bodies and the eccentricity of theirelliptic orbits. The eccentricity is assumed to be small. Nonlinear equations of motion ofthe test mass near a triangular libration point are analyzed. It is assumed that theparameters of the problem lie on the curves of third-order resonances corresponding tothe planar restricted problem.In addition to these resonances (their number is equal to five), the spatial problemhas a resonance that takes place at any parameter values since thethe frequency of small linear oscillations of the test mass along the axis perpendicular tothe plane of the orbit of the main bodies is equal to the frequency of Keplerian motion ofthese bodies.In this paper, the normal form of the Hamiltonian function of perturbed motionthrough fourth-degree terms relative to deviations from the libration point is obtained.Explicit expressions for the coefficients of normal form up to and including the seconddegree of eccentricity are found.

The motion of a heavy rigid body with suspension point performing high-frequency periodic vibrations of small amplitude is considered. The study is carried out within the framework of an approximate autonomous system written in the form of the modified Euler – Poisson equations, to the right-hand sides of which the components of the vibration moment are added. The question of the existence of two particular motions of the body is resolved, they are permanent rotations and pendulum-type motions. It is shown that permanent rotations of the body can occur in the case of vibration symmetry relative to a vertically located axis. The search for pendulum-type motions is restricted to the case when the axis of these motions is one of the principal inertia axes of the body, as in the case of a heavy rigid body with a fixed point. Two basic variants of vibrations are considered, when the suspension point vibrates along a straight line and along an ellipse. To the latter variant any planar vibrations and a wide class of spatial vibrations of the suspension point are reduced.It is shown that for both basic cases of vibrations, pendulum-type motions are of two types. The motions of the first type are similar to the Mlodzeevsky’s pendulum-type motions of a heavy rigid body with a fixed point. For them, the body’s mass center lies in the principal plane of inertia, and the axis of the pendulum-type motions is perpendicular to this plane. Pendulum-type motions of the second type occur around the principal axis of inertia containing the body’s center of mass. Such motions are absent in the gravitational problem, they are caused by the presence of vibrations. To search for the pendulum-type motions, an approach is proposed that combines theresults of the problem of gravitation (without vibration) and that of vibration(ignoring gravity). As an illustration, a number of examples of the interaction of the gravitational field and the vibration field corresponding to both basic variants of vibrations of the body’s suspension point are considered.

For an autonomous dynamical system of (n) differential equations each integral of motion allows for reduction of the order of equations by 1 and knowledge of ((n-1)) integrals is necessary for the system to be integrated by quadratures. The amenability of Hamiltonian systems to being integrated by quadratures is characterised by the Liouville theorem where in (2n)-dimensional phase space only (n) integrals are sufficient as equations are generated by 1 function — the Hamiltonian.

There are, however, large families of Newton-type differential equations for which knowledge of 2 or 1 integral is sufficient for recovering separability and integration by quadratures. The purpose of this paper is to discuss a tradeoff between the number of integrals and the special structure of autonomous, velocity-independent 2nd order Newton equations (ddot{boldsymbol{q}}=boldsymbol{M}(boldsymbol{q})), (boldsymbol{q}inmathbb{R}^{n}), which allows for integration by quadratures.

In particular, we review little-known results on quasipotential and triangular Newton equations to explain how it is possible that 2 or 1 integral is sufficient. The theory of these Newton equations provides new types of separation webs consisting of quadratic (but not orthogonal) surfaces.

We prove the integrability of magnetic geodesic flows of (SO(n))-invariant Riemannian metrics on the rank two Stefel variety (V_{n,2}) with respect to the magnetic field (eta dalpha), where (alpha) is the standard contact form on (V_{n,2}) and (eta) is a real parameter.Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for (SO(n))-invariant sub-Riemannian structures on (V_{n,2}). All statements in the limit (eta=0) imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by (SO(n)times SO(2))-invariant Riemannian metrics. For (n=3), using the isomorphism (V_{3,2}cong SO(3)), the obtained integrable magnetic models reduce tointegrable cases of the motion of a heavy rigid body with a gyrostat around a fixed point:the Zhukovskiy – Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevskitop with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrangegyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).

This paper provides an original rendition of the heavy top that unravels the mysteries behind S. Kowalewski’s seminal work on the motions of a rigid body around a fixed point under the influence of gravity.The point of departure for understanding Kowalewski’s workbegins with Kirchhoff’s model for the equilibrium configurations of an elastic rod in ({mathbb{R}}^{3}) subject to fixed bending and twisting moments at its ends [17]. This initial orientation to the elastic problem shows, first, that the Kowalewski type integrals discovered by I. V. Komarov and V. B. Kuznetsov [24, 25] appear naturally on the Lie algebras associated with the orthonormal frame bundles of the sphere (S^{3}) and the hyperboloid (H^{3}) [17] and, secondly, it showsthat these integrals of motion can be naturally extracted from a canonical Poisson system on the dual of (so(4,mathbb{C})) generated byan affine quadratic Hamiltonian (H) (Kirchhoff – Kowalewski type).

The paper shows that the passage to complex variablesis synonymous with the representation of (so(4,mathbb{C})) as (sl(2,mathbb{C})times sl(2,mathbb{C})) and the embedding of (H) into (sp(4,mathbb{C})), an important intermediate step towards uncovering the origins of Kowalewski’s integral. There is a quintessential Kowalewski type integral of motion on (sp(4,mathbb{C})) that appears as a spectral invariant for the Poisson system associated with a Hamiltonian (mathcal{H}) (a natural extension of (H)) that satisfies Kowalewski’s conditions.

The text then demonstrates the relevance of this integral of motion for other studies in the existing literature [7, 35]. The text also includes a self-contained treatment of the integration of the Kowalewski type equations based on Kowalewski’s ingenuous separation of variables, the hyperelliptic curve and the solutions on its Jacobian variety.