Regular and Chaotic Dynamics最新文献

The triple linkage of Thurston and Weeks exhibits Anosov behavior for certain parameter values,which can be shown by examining the Gauss curvature of the configuration space equipped with the metric induced by kinetic energy.In this paper, we consider a spatial linkage that can be viewed as a conversion of the triple linkage.We show that the configuration space asymptotically becomes a Riemannian submanifold of the four-dimensional torus (mathbb{T}^{4}) taking the limit of the parameters.Through verified numerical computation, we demonstrate that the asymptotic configuration space has negative curvature,and hence that for parameters close to the limit the linkage is Anosov by structural stability of an Anosov flow.

The problem of the motion of a dynamically symmetric ball with a shifted center of gravity on a horizontal absolutely rough plane is considered, taking into account the unilateral nature of the contact. It is shown that for some values of the parameters and initial conditions, singularities of two types may appear: ambiguity in determining the subsequent motion (separation of the body from the support or continuation of rolling) or the impossibility of determining it within the framework of the model used (the finite time paradox). For a formal description of these situations, the generalized complementarity problem is used. The Littlewood problem of the rolling of a hoop with a point mass is studied in detail. Some cases of paradoxes when imposing differential constraints of various types are also discussed, including the generalization of the Chaplygin sleigh and the “rubber” ball.

We consider a new type of billiard trajectories of point-particles moving freely in (d)-dimensional space until collision. At collisions of two or more particles we have scattering at a subspace with a co-dimension of at least (d) preserving only the total momentum of the colliding particles, but the internal direction and kinetic energy can change arbitrary and even an exchange of mass is possible. Hence the future of the trajectory is nondeterministic. Motivated by questions concerning non-collision singularities in the (n)-body problem, for which these systems might serve as approximations, we are mainly interested in the asymptotic growth rate for trajectories that have infinitely many collisions and are expanding. For this case we provide as our main results exponential lower bounds for the diameter and the kinetic energy of the system in the number of so-called chain-closing collisions.

For any given positive masses, we prove that the number of (mathbf{S})-balanced configurations of four bodies in the plane is finite up to similitudes, provided that the symmetric matrix (mathbf{S}) is sufficiently close to a numerical matrix. To establish this result, we utilize singular sequences to analyze the possible degenerate algebraic varieties defined by (mathbf{S})-balanced configurations. We derive all potential singular diagrams, encompassing both equal-order and non-equal-order cases. In the equal-order case, we obtain the necessary mass equations, while for (mathbf{S}) approaching the identity matrix, we demonstrate the absence of non-equal-order singular sequences, thereby rigorously rule out all non-generic scenarios. Furthermore, we extend this conclusion to the five-body scenario.

A caustic of a billiard is a curve whose tangent lines are reflected to its own tangent lines. A billiard is called Birkhoff caustic-integrable if there exists a topological annulus adjacent to its boundary from inside that is foliated byclosed caustics. The famous Birkhoff Conjecture, studied by many mathematicians, states that the only Birkhoff caustic-integrable billiards are ellipses. The conjecture is open even for billiards whose boundaries are ovals of algebraic curves. In this case the billiard is known to have a dense family of so-called rational caustics that are also ovals of algebraic curves.We introduce the notion of acomplex caustic: a complex algebraic curve whose complex tangent lines are sent by complexified reflection to its owncomplex tangent lines.We show that the usual billiard on a real planar curve (gamma) has a complex caustic if and only if (gamma) is a conic. We prove an analogous result for billiards on all the surfaces of constant curvature. These results are corollaries of the solution of S. Bolotin’s polynomial integrability conjecture: a joint result by M. Bialy, A. Mironov and the author. We extend themto the projective billiards introduced by S. Tabachnikov, which are a common generalization of billiards on surfaces of constant curvature. We also deal with a well-known class of projective billiards on conics that are defined to have caustics forming a dual conical pencil. We show that, up to restriction to a finite union of arcs, each of them is equivalent to a billiard on an appropriate surface of constant curvature.

In this paper we investigate rhomboidal central configurations for the planar five-body problem.These are configurations whose convex hulls are rhombuses, and one interior mass lies on a diagonal of the rhombus.Consider partition of the configuration space by orderings of mutual distances,our main theorem provides criteria for central configurations in every region of this partition.For some regions we obtain existence and uniqueness of central configurations, for some we obtainnonexistence, for others our criteria are necessary conditions on masses.Our proofs are elementary and completely analytic.

The Burgers – Gardner equation with a nonlinear source is considered. The Cauchy problem for this equation cannot be solved by the inverse scattering transform and we look for the traveling wave solutions. The Painlevé test is used to find constraints on the parameters of the equation for its integrability. The first integral of the nonlinear ordinary differential equation is obtained taking into account the results of thePainlevé test. The first integral thus found is used to obtain the general solutionof the equation. Some exact solutions are found by means of the simplest equation method.