Universal homoclinic bifurcations and chaos near double resonances
This paper provides a careful study of the near integrable Hamiltonian system with Hamiltonian \(H(I,\varphi, \varepsilon)= H_0(I)+ \varepsilon H_1(I, \varphi, \varepsilon)\) near some resonant \(n\)-torus \(\{I= I^r\}\). It is assumed that for the unperturbed system at \(\varepsilon=0\) exactly two independent resonance conditions of the form \(\langle D_IH_0 (I^r),k \rangle =0\) hold and that one of these resonances is of much higher order than the other, measured by some parameter \(\mu\). In this case one can derive a normal form near the invariant torus containing the parameter \(\mu\). For a truncated normal form the existence of invariant hyperbolic \((n-2)\)-tori is established. The invariant manifolds of these tori intersect each other in different types of multipulse-homoclinic orbits. It is a remarkable result that a universal bifurcation diagram for these homoclinic bifurcations can be given that is independent of the concrete form of the Hamiltonian. After discussing persistence of the homoclinic orbits in the full normal form, the author applies his results to a three-degree-of-freedom mechanical system consisting of three coaxial disks.
scientific article; zbMATH DE number 1279766