# Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio

*2014, Volume 19, Number 6, pp. 663-680*

Author(s):

**Delshams A., Gonchenko M. S., Gutierrez P.**

We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega = \sqrt{2}−1$. We show that the Poincaré–Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the transversality of the splitting whose dependence on the perturbation parameter ε satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.

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