Degenerate Bifurcations of Resonant Tori in Hamiltonian Systems

Abstract

The twist condition is a necessary condition in integrable Hamiltonian systems and symplectic maps to obtain Poincaré–Birkhoff bifurcations under small perturbations. When this condition does not hold, topological structures other than Poincaré–Birkhoff chains arise in phase space through bifurcations of isolated periodic orbits and reconnections of asymptotic manifolds. In this paper we construct an integrable model Hamiltonian with degeneracies suitable to observe these phenomena close to nontwist resonant tori. The generation of isochronous chains and the stability of their fixed points is determined analytically and a condition for the reconnection is found. Particular examples are given, illustrating the bifurcation and reconnection scenario for several cases of degeneracy.

Large-scale chaos for arbitrarily small perturbations in non-twist Hamiltonian systems

“…The functions f, F are assumed to be analytic functions of I 1 andā denotes a parameter vector. In [9] F (I 1 ) is assumed to be a polynomial function which corresponds to the Birkhoff normal form. This is not necessary but, on the other hand, it does not affect the genericity of the study.…”

“…For a critical value of ∆I (i) 1 (for some parameter valuesā =ā rec and k = k rec ) the homoclinic manifolds of the isochronous chains are connected smoothly (separatrix reconnection). For the system (5) the reconnection condition is [9] …”

“…The stability of the 2rm fixed points is studied in [9] and the results obtained there can be applied directly to system (9). When condition (7) is satisfied then m isochronous Poincaré-Birkhoff chains appear and they are assigned by their particular sets of fixed points…”

“…For chains of "Type I" the fixed points, P (i, j) and P (i + 1, j) are of different stability for all j ≤ r, while, for chains of "Type II", P (i, j) and P (i + 1, j) are of the same stability type for all j ≤ r. In [9] it is shown that when the restriction…”

“…In section 2 we summarize and discuss some results obtained in [9] that refer to an integrable nontwist Hamiltonian model. In section 3, we show that nontwist phenomena can be studied by considering one degree of freedom systems with periodic-time dependent perturbations.…”