69 research outputs found
Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results
We study the problem of exponentially small splitting of separatrices of one
degree of freedom classical Hamiltonian systems with a non-autonomous
perturbation fast and periodic in time. We provide a result valid for general
systems which are algebraic or trigonometric polynomials in the state
variables. It consists on obtaining a rigorous proof of the asymptotic formula
for the measure of the splitting. We obtain that the splitting has the
asymptotic behavior ,
identifying the constants in terms of the system features.
We consider several cases. In some cases, assuming the perturbation is small
enough, the values of coincide with the classical Melnikov approach.
We identify the limit size of the perturbation for which this theory holds
true. However for the limit cases, which appear naturally both in averaging and
bifurcation theories, we encounter that, generically, and are not
well predicted by Melnikov theory
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