Transversality of homoclinic orbits to hyberbolic equilibria in a Hamiltonian system, via the Hamilton--Jacobi equation

摘要

We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolicequilibrium point having a loop or homoclinic orbit (or, alternatively, twohyperbolic equilibrium points connected by a heteroclinic orbit), as a steptowards understanding the behavior of nearly-integrable Hamiltonians neardouble resonances. We provide a constructive approach to study whether theunstable and stable invariant manifolds of the hyperbolic point intersecttransversely along the loop, inside their common energy level. For the systemconsidered, we establish a necessary and sufficient condition for thetransversality, in terms of a Riccati equation whose solutions give the slopeof the invariant manifolds in a direction transverse to the loop. The key pointof our approach is to write the invariant manifolds in terms of generatingfunctions, which are solutions of the Hamilton--Jacobi equation. In someexamples, we show that it is enough to analyse the phase portrait of theRiccati equation without solving it explicitly. Finally, we consider ananalogous problem in a perturbative situation. If the invariant manifolds ofthe unperturbed loop coincide, we have a problem of splitting of separatrices.In this case, the Riccati equation is replaced by a Mel'nikov potential definedas an integral, providing a condition for the existence of a perturbed loop andits transversality. This is also illustrated with a concrete example.