Asymptotic Expansions of Unstable and Stable Manifolds in Time-Discrete Systems

Abstract

By means of an updated renormalization method, we construct asymptotic expansions for unstable manifolds of hyperbolic fixed points in the double-well map and the dissipative Hénon map, both of which exhibit the strong homoclinic chaos. In terms of the asymptotic expansion , a simple formulation is presented to give the first homoclinic point in the double-well map. Even a truncated expansion of the unstable manifold is shown to reproduce the well-known many-leaved (fractal) structure of the strange attractor in the Hénon map.

Random Wandering around Homoclinic-Like Manifolds in a Symplectic Map Chain

“…In Fig. 1 , trajectories obtained from the naive RG map (8) and the regularized RG map ( 14) are depicted to be compared to an exact trajectory of the original map (Eqs. ( 1)-( 2)).…”

“…However, the conventional asymptotic methods such as the averaging method or the method of the multiple-time scales may not be immediately applicable to discrete systems. Instead, we employ the perturbative renormalization group (RG) method [6] [7], which can apply to a discrete system and leads to a reduced map as the RG map [8]. However, a naive reduced map does not preserve symplectic symmetry and fails to describe long time behavior of the original symplectic map.…”

Renormalization Analysis of Resonance Structure in a 2-D Symplectic Map

“…Since the renormalization group (RG) method was developed as an asymtotic method to study a long-time behaviour of a flow in a dynamical system, [1] there have been several attempts to extend the method to apply to discrete systems. [5,3] Particularly, the application to a symplectic map is of great importance since a symplectic map is generally derived as a Poincare map of a Hamiltonian flow describing a physical system. However, a naive application of the RG method fails to describe a long-time behaviour of a system since the naive renormalization process does not secure the symplectic smmmetry to an RG map [4] .…”

Liouville operator approach to symplecticity-preserving renormalization group method

“…The RG method is reformulated on the basis of a naive renormalization transformation and the Lie group [4]. This reformulated RG method based on the Lie group is easy to apply to discrete systems, by which asymptotic expansions of unstable manifolds of some chaotic discrete systems are obtained [5]. The extension of the RG method to discrete symplectic systems is not trivial because the symplectic struture is easily broken in naive RG equations (maps) as shown in Ref.…”