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Exponentially small splitting of the rapidly forced pendulum via Hamilton-Jacobi equation and Resurgence: a proof of the singular case
by
Marcel Guardia
Universitat Politecnica de Catalunya
Coauthors: Tere M. Seara and Carme Olive
In this work is studied the separatrix splitting for the rapidly periodically forced pendulum
d2x/dt2=sinx+(m/e2)sin(t/e)
where e is a small positive real parameter and m < m0 where m0 is the first zero of the Bessel function of order zero
J0(m)=(1/2p)∫02p cos(msint)dt
It is obtained a rigourous proof of the asymptotic expression of the exponentially small splitting of separatrices. It is seen that for m ≤ O(eq) with q > -3/2 Melnikov theory is valid and that for m ≤ O(eq) with q ∈ (-2, -3/2) is also valid doing previously two steps of averaging which modify the non-perturbed system. For the limit case m = 0(1), it is proved that the Melnikov function fails to predict the asymptotic expression of distance between manifolds. Moreover, the splitting of separatrices is computed in the original variables.
In the proof, the invariant manifolds are studied as solutions of the Hamilton-Jacobi equation as suggested in [1]. Moreover, an auxiliar system defined in [2] as a reference system is studied with Ecalle Resurgence Theory, as in [3], which lead to the asymptotic expression of the splitting of separatrices. Finally, complex matching techniques complete the proof.
[1] D. Sauzin, A new method for measuring the splitting of invariant manifolds, Annales Scientifiques de l'Ecole Normale Superieure (2001)
[2] V. Gelfreich, Reference systems for splitting of separatrices, Nonlinearity (1997)
[3] C. Olive, D. Sauzin, T. M. Seara, Resurgence in a Hamilton-Jacobi equation, Proceedings of the International Conference in Honor of Frederic Pham (Nice, 2002), Ann. Inst. Fourier (2003)
Date received: June 15, 2007
Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavg-02.