Abstract
Using stationary phase methods, we provide an explicit formula for the Melnikov function of the one and a half degrees of freedom system given by a Hamiltonian system subject to a rapidly oscillating perturbation. Remarkably, the Melnikov function turns out to be computable using very little information on the separatrix and in the case of non-analytic systems. This is related to a priori stable systems coupled with low regularity perturbations. A natural physical application is to perturbations controlled by wave-type equations, so in particular we also illustrate this result with the motion of charged particles in a rapidly oscillating electromagnetic field. Quasi-periodic perturbations are discussed too.
Similar content being viewed by others
References
Baldomá I., Fontich E., Guardia M., Seara T.M.: Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results. J. Differ. Equ. 253, 3304–3439 (2012)
Belitskii, G., Tkachenko, V.: One-dimensional functional equations, Birkhäuser, Basel (2003)
Burns K., Weiss H.: A geometric criterion for positive topological entropy. Commun. Math. Phys. 172, 95–118 (1995)
Chierchia L., Gallavotti G.: Drift and diffusion in phase space. Ann. Inst. H. Poincaré 60, 1–144 (1994)
De Lellis C., Székelyhidi L.: High dimensionality and h-principle in PDE. Bull. Am. Math. Soc. 54, 247–282 (2017)
Delshams A., Gelfreich V., Jorba A., Seara T.M.: Exponentially small splitting of separatrices under fast quasiperiodic forcing. Commun. Math. Phys. 189, 35–71 (1997)
Delshams A., Gonchenko M., Gutiérrez P.: Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio. Regul. Chaotic Dyn. 19, 663–680 (2014)
Delshams A., Gonchenko M., Gutiérrez P.: Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio. SIAM J. Appl. Dyn. Syst. 15, 981–1024 (2016)
Delshams A., Seara T.M.: An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Commun. Math. Phys. 150, 433–463 (1992)
Eliashberg, Y., Mishachev, N.: Introduction to the h-principle, American Mathematical Society, Providence (2002)
Enciso A., Lucà R., Peralta-Salas D.: Vortex reconnection in the three dimensional Navier–Stokes equations. Adv. Math. 309, 452–486 (2017)
Enciso, A., Peralta-Salas, D.: A problem of Ulam about magnetic fields generated by knotted wires. Ergodic Theory Dyn. Syst. (2017). https://doi.org/10.1017/etds.2017.117
Féjoz J., Guardia M., Kaloshin V., Roldán P.: Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem. J. Eur. Math. Soc. 18, 2315–2403 (2016)
Fontich E.: Rapidly forced planar vector fields and splitting of separatrices. J. Differ. Equ. 119, 310–335 (1995)
Grafakos, L.: Classical Fourier Analysis, Springer, New York (2014)
Guardia M., Martí n P., Seara T.M.: Oscillatory motions for the restricted planar circular three body problem. Invent. Math. 203, 417–492 (2016)
Guardia M., Seara T.M.: Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation. Nonlinearity 25, 1367–1412 (2012)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1990)
Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds, Springer, New York (1977)
Katok A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. IHES 51, 137–173 (1980)
Lochak P., Marco J.-P., Sauzin D.: On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems. Mem. Am. Math. Soc. 163, 1–145 (2003)
Lombardi, E.: Oscillatory Integrals and Phenomena Beyond All Algebraic Orders, Springer, New York (2000)
Nash J.: C 1 isometric imbeddings. Ann. Math. 60, 383–396 (1954)
Nazarov F., Sodin M.: On the number of nodal domains of random spherical harmonics. Am. J. Math. 131, 1337–1357 (2009)
Sauzin D.: Résurgence paramétrique et exponentielle petitesse de l’écart des séparatrices du pendule rapidement forcé. Ann. Ins. Fourier 45, 453–511 (1995)
Sauzin D.: A new method for measuring the splitting of invariant manifolds. Ann. Sci. Éc. Norm. Sup. 34, 159–221 (2001)
Smale S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)
Stein, E.M.: Harmonic Analysis, Princeton University Press, Princeton (1993)
Acknowledgments
A.E. and D.P.-S. are respectively supported by the ERCStartingGrants 633152 and 335079. A.L. is supported by the Knut och Alice Wallenbergs stiftelse KAW 2015.0365. This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Liverani
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Enciso, A., Luque, A. & Peralta-Salas, D. Stationary Phase Methods and the Splitting of Separatrices. Commun. Math. Phys. 368, 1297–1322 (2019). https://doi.org/10.1007/s00220-019-03364-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-019-03364-0