Outer Billiards outside a Regular Octagon: Periodicity of Almost All Orbits and Existence of an Aperiodic Orbit

The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.     

Billiards and Nonholonomic Distributions

In this note, we consider billiards with full families of periodic orbits. It is shown that the construction of a convex billiard with a “rational” caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to an (N - 1)-dimensional distribution on a (2N - 1)-dimensional manifold. We describe the properties of this distribution, as well as some important consequences for billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist. Bibliography: 13 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 56–64.     

Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction

Summary We have applied topological methods to analyze chaotic time series data from the Belousov-Zhabotinskii reaction. First, the periodic orbits shadowed by the data set were identified. Next, a three-dimensional embedding without self-intersections was constructed from the data set. The topological structure of that flow was visualized by constructing a branched manifold such that every periodic orbit in the flow could be held by the branched manifold. The branched manifold, or induced template, was computed using the three lowest-period orbits. The organization of the higher-period orbits predicted by this induced template was compared with the organization of the orbits reconstructed from the data set with excellent results. The consequences of the presence of certain knots found in the data are discussed.     

Code and order in polygonal billiards

Two polygons P, Q are code equivalent if there are billiard orbits u, v which hit the same sequence of sides and such that the projections of the orbits are dense in the boundaries ∂P, ∂Q. Our main results show when code equivalent polygons have the same angles, resp. are similar, resp. affinely similar.     

Periodic billiard paths in right triangles are unstable

A stable periodic billiard path in a triangle is a billiard path which persists under small perturbations of the triangle. This article gives a geometric proof that no right triangles have stable periodic billiard paths.      

Lyapunov orbits in the n-vortex problem

In the reduced phase space by rotation, we prove the existence of periodic orbits of the n-vortex problem emanating from a relative equilibrium formed by n unit vortices at the vertices of a regular polygon, both in the plane and at a fixed latitude when the ideal fluid moves on the surface of a sphere. In the case of a plane we also prove the existence of such periodic orbits in the (n + 1)-vortex problem, where an additional central vortex of intensity κ is added to the ring of the polygonal configuration.     

Systematic Computer Assisted Proofs of periodic orbits of Hamiltonian systems

The numerical study of Dynamical Systems leads to obtain invariant objects of the systems such as periodic orbits, invariant tori, attractors and so on, that helps to the global understanding of the problem. In this paper we focus on the rigorous computation of periodic orbits and their distribution on the phase space, which configures the so called skeleton of the system. We use Computer Assisted Proof techniques to make a rigorous proof of the existence and the stability of families of periodic orbits in two-degrees of freedom Hamiltonian systems, which provide rigorous skeletons of periodic orbits. To that goal we show how to prove the existence and stability of a huge set of discrete initial conditions of periodic orbits, and later, how to prove the existence and stability of continuous families of periodic orbits. We illustrate the approach with two paradigmatic problems: the Hénon–Heiles Hamiltonian and the Diamagnetic Kepler problem.     

Shortest periodic billiard trajectories in convex bodies

We show that the length of any periodic billiard trajectory in any convex body is always at least 4 times the inradius of K; the equality holds precisely when the width of K is twice its inradius, e.g., K is centrally symmetric, in which case we prove that the shortest periodic trajectories are all bouncing ball (2-link) orbits.     

Simulated annealing algorithm for finding periodic orbits of multi-electron atomic systems

We adapt the simulated annealing algorithm to the search of periodic orbits for classical multi-electron atomic systems. This is done by minimizing the nth return distance to the initial position on a Poincaré surface of section under an energy constraint. Here we give evidence of the feasibility of the method by applying it to the helium atom in the ground state for one to three spatial dimensions. We examine the structure of the dynamics and connect its organization to the periodic orbits we have found.     

On topological entropy of billiard tables with small inner scatterers

We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik-Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small.     

Orbits in max-min algebra

Properties of orbits in max-min algebra are described, mainly the properties of periodic orbits. An O(n³) algorithm computing the period of a periodic orbit is presented. As a consequence, an O(n³ log n) algorithm computing the period of arbitrary orbit is obtained, as the pre-periodic part of the orbit has length at most (n − 1)² + 1.     

EQUIDISTRIBUTION OF CLOSED ORBITS OF BILLIARD FLOWS ON THE PLANE

The authors consider the billiard system with finitely many convex scatters with smooth boundary satisfying the visibility assumption on the plane and prove that the closed orbits for the billiard flow is uniformly distributed.     

Point-to-Periodic and Periodic-to-Periodic Connections

In this work we consider computing and continuing connecting orbits in parameter dependent dynamical systems. We give details of algorithms for computing connections between equilibria and periodic orbits, and between periodic orbits. The theoretical foundation for these techniques is given by the seminal work of Beyn in 1994, “On well-posed problems for connecting orbits in dynamical systems”, where a numerical technique is also proposed. Our algorithms consist of splitting the computation of the connection from that of the periodic orbit(s). To set up appropriate boundary conditions, we follow the algorithmic approach used by Demmel, Dieci, and Friedman, for the case of connecting orbits between equilibria, and we construct and exploit the smooth block Schur decomposition of the monodromy matrices associated to the periodic orbits. Numerical examples illustrate the performance of the algorithms. This revised version was published online in July 2006 with corrections to the Cover Date.     

Polygons in billiard orbits

We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these polygons.     

Homoclinic orbits to a saddle-center in a fourth-order differential equation

We study homoclinic orbits to a saddle-center of a fourth-order ordinary differential equation, which is invariant under the transformation x→−x, involving an eigenvalue parameter q and an odd, piece-wise, cubic-type nonlinearity. It is found that for a sequence of eigenvalues which tends to infinity, homoclinic orbits exist whose complexity increases as the eigenvalue becomes larger. These orbits are found to be embedded in branches of homoclinic orbits to periodic orbits as x→±∞.     

Discrete wavelets and the Vilenkin-Chrestenson transform

In the spaces of complex periodic sequences, we use the Vilenkin-Chrestenson transforms to construct new orthogonal wavelet bases defined by finite collections of parameters. Earlier similar bases were defined for the Cantor and Vilenkin groups by means of generalized Walsh functions. It is noted that similar constructions can be realized for biorthogonal wavelets as well as for the space ? ²(?₊).     

一类弥散台球流的闭轨道分布

考虑三维空间中满足可视条件的、具有光滑边界的有限个散射体组成的区域中的台球流,给出了具有碰撞数为偶数的闭轨道条数的渐近分布.     

On 4-Reflective Complex Analytic Planar Billiards

The famous conjecture of Ivrii (Funct Anal Appl 14(2):98–106, 1980) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the author’s previous result Glutsyuk (Moscow Math J 14(2):239–289, 2014) classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar \(C^4\)-smooth pseudo-billiards; solutions of Tabachnikov’s Commuting Billiard Conjecture and the 4-reflective case of Plakhov’s Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise \(C^4\)-smooth). We provide a survey and a small technical result concerning higher number of complex reflections.     

具有共振特征值的轨道翻转双同宿环分支

该文研究了具有轨道翻转的双同宿环四维系统,在主特征值共振和沿轨道奇点处切方向共振下的两种分支．我们分别在系统奇点小邻域内利用规范型的解构造一个奇异映射,再在双同宿环的管状邻域内引起局部活动坐标架,利用系统线性变分方程的解定义了一个正则映射,通过复合两个映射而得到分支研究中一类重要的Poincaré映射,经过简单的计算最终得到后继函数的精确表达式．对分支方程细致地研究,我们给出了原双同宿环的保存性条件,并证明了“大” 1-同宿环分支曲面,2-重“大”1-周期轨分支曲面,“大”2-同宿环分支曲面的存在性、存在区域和近似表达式,及其分支出的“大”周期轨和“大”同宿轨的存在性区域和数量．     

An example on movable approximations of a minimal set in a continuous flow

In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R³, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the paper consists of constructing an example of a set in R³ that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question.