Yi-Chiuan Chen

Anti-integrability in scattering billiards

Examples that scattering billiards exhibit anti-integrability are shown. A Cantor set can be constructed such that the generalized Sinai billiard map under consideration restricted to it is conjugate to a subshift of finite type with a given number of symbols. The chaotic and Bernoulli properties are inherited from the ‘δ-billiards’.

Abrupt bifurcations in chaotic scattering: view from the anti-integrable limit*

Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height Ec there is a hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > Ec there are no bounded orbits. They called the bifurcation at E = Ec an abrupt bifurcation to chaotic scattering.The aim of this paper is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the anti-integrable limit, and to do so for a general range of chaotic scattering problems.

FAMILY OF INVARIANT CANTOR SETS AS ORBITS OF DIFFERENTIAL EQUATIONS

The invariant Cantor sets of the logistic map gμ(x) = μx(1 - x) for μ > 4 are hyperbolic and form a continuous family. We show that this family can be obtained explicitly through solutions of initial value problems for a system of infinitely coupled differential equations due to the hyperbolicity. The same result also applies to the tent map Ta(x) = a(1/2 - |1/2 - x|) for a > 2.

CHAOTIC ORBITS IN A PLANAR THREE-CENTER PROBLEM OF SLIGHTLY NEGATIVE ENERGY

The orbital fates in a planar three-center problem of slightly negative energy are studied numerically, by classifying the dynamics into colliding with one of the centers, escaping to infinity and orbiting around the centers. We consider three energy levels, for each of them, the set of initial values leading to collision with the centers is determined. It presents an intriguing fractal structure. The complementary set, which corresponds to those initial values whose orbits are either regular or chaotic winding around the fixed centers, also exhibits a fractal structure. These fractal structures found here might lead to some observable physical feature in the future. The fractal structures imply the sensitivity to initial conditions, thus enable us to find initial values with which the orbits are chaotic, having a positive Lyapunov exponent.

FAMILY OF INVARIANT CANTOR SETS AS ORBITS OF DIFFERENTIAL EQUATIONS II: JULIA SETS

The Julia set of the quadratic map fμ(z) = μz(1 - z) for μ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely coupled differential equations. A key point is that the required initial values can be obtained from the anti-integrable limit μ → ∞. The system of infinitely coupled differential equations reduces to a finitely coupled one if we are only concerned with some invariant finite subset of the Julia set. Therefore, it can be employed to find periodic orbits as well. We conduct numerical approximations to the Julia sets when parameter μ is located at the Misiurewicz points with external angle 1/2, 1/6, or 5/12. We approximate these Julia sets by their invariant finite subsets that are integrated along the reciprocal of corresponding external rays of the Mandelbrot set starting from the anti-integrable limit μ = ∞. When μ is at the Misiurewicz point of angle 1/128, a 98-period orbit of prescribed itinerary obtained by this method is presented, without having to find a root of a 298-degree polynomial. The Julia sets (or their subsets) obtained are independent of integral curves, but in order to make sure that the integral curves are contained in the exterior of the Mandelbrot set, we use the external rays of the Mandelbrot set as integral curves. Two ways of obtaining the external rays are discussed, one based on the series expansion (the Jungreis–Ewing–Schober algorithm), the other based on Newtons method (the OTIS algorithm). We establish tables comparing the values of some Misiurewicz points of small denominators obtained by these two algorithms with the theoretical values.

Family of Smale–Williams Solenoid Attractors as Solutions of Differential Equations: Exact Formula and Conjugacy

We show that the family of the Smale–Williams solenoid attractors parameterized by its contraction rate can be characterized as solutions of a set of differential equations. The exact formula describing the attractor can be obtained by solving the differential equations subject to explicitly given initial conditions. Using the formula, we present in this note a simple and explicit proof of the result that the dynamics on the solenoid is topologically conjugate to the shift on the inverse limit space of the expanding map t ↦ mt mod 1 for some integer m ≥ 2 and to a suspension over the adding machine.

Topological horseshoes in travelling waves of discretized nonlinear wave equations

Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.