Abstract
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.
Highlights
Bleher et al [7] found numerically an interesting chaotic phenomenon in 1989 when investigating the motion of a particle scattered by a smooth planar potential field V : R2 → R
When the total energy E of the particle is slightly less than the height Ec := max(x,y)∈R2 V (x, y) of the peaks of the potential, their study suggested that there exists a bounded hyperbolic invariant set of the form of a suspension of a Cantor set in the energy level, on which chaotic scattering occurs, whereas there are no bounded orbits when E > Ec
Ec with E < Ec, they proposed that the Lyapunov exponents of orbits of the return map to a cross-section of the invariant set are of order ln(Ec − E)−1, while the box-counting dimension of a cross-section to the invariant set is asymptotically proportional to 1/ ln(Ec − E)−1
Summary
Bleher et al [7] found numerically an interesting chaotic phenomenon in 1989 when investigating the motion of a particle scattered by a smooth planar potential field V : R2 → R of the form. In our paper we develop a general approach to proving chaotic scattering, in particular constructing hyperbolic suspensions of topological Markov chains. It still does not prove the full extent of the claims of [6], for (1) because of the right angles between the heteroclinic orbits along the edges of the square (a problem already recognized in [6]), and for (2) because the hills are elliptic with their short axes pointing towards the centre whereas we would need the long axes towards the centre to deduce abrupt formation of a topological.
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