Renormalization Theory for Hamiltonian Systems

Abstract

We study the onset of widespread chaos in Hamiltonian systems with two degrees of freedom. Such systems and their stability properties are of interest in diverse fields (celestial mechanics, plasma physics, chemical physics to name just a few). Due to topological reasons, two-dimensional invariant tori of irrational winding numbers represent barriers to widespread chaos. The breakup of the "last" invariant torus can be viewed as the threshold to widespread chaos. We use the renormalization group approach in order to describe the breakup of invariant tori of irrational winding numbers. An approximate renormalization scheme is derived for this purpose.The scheme is implemented with the help of the "Maple" computer algebra system. The renormalization group approach is applied to a number of systems. We discuss the paradigm Hamiltonian of Escande and Doveil, the Walker and Fordmodel, a model of the ethane molecule, the double pendulum, the Baggott system, limacon billiards. The Poincare surface of section technique is used in order to study numerically the dynamic behavior of the systems and to check the results of the renormalization theory.