Global Effect of Non-Conservative Perturbations on Homoclinic Orbits

• Gidea Marian ^[1] ; de la Llave Rafael ^[2] ; Musser, Maxwell ^[1]
  1. [1] Yeshiva University

     Yeshiva University

     Estados Unidos

     
  2. [2] Georgia Institute of Technology

     Georgia Institute of Technology

     Estados Unidos

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 1, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-020-00431-z
• Enlaces
  • Texto completo
• Resumen

  • We study the effect of time-dependent, non-conservative perturbations on the dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. We assume that the unperturbed system is Hamiltonian, and the normally hyperbolic invariant manifold is parametrized via action-angle coordinates. The homoclinic excursions can be described via the scattering map, which gives the future asymptotic of an orbit as a function of its past asymptotic. We provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map expressed in action-angle coordinates. We illustrate these formulas for perturbations of both uncoupled and coupled rotator-pendulum systems.

• Referencias bibliográficas
  • 1. Baldoma, M., Fontich, E.: Poincaré-Melnikov theory for n-dimensional diffeomorphisms. Appl Math 25(2), 129–152 (1998)
  • 2. Burns, Keith, Gidea, Marian: Differential Geometry and Topology: With a View to Dynamical Systems. Studies in Advanced Mathematics. Chapman...
  • 3. del Castillo-Negrete, D., Morrison, P.J.: Hamiltonian chaos and transport in quasigeostrophic flows. Chaotic dynamics and transport in...
  • 4. Delshams, Amadeu, de la Llave, Rafael, Seara, Tere M.: A geometric approach to the existence of orbits with unbounded energy in generic...
  • 5. Delshams, A., De la Llave, R. and Seara, T.M.: A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem:...
  • 6. Delshams, Amadeu, de la Llave, Rafael, Seara, Tere M.: Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows. Adv....
  • 7. Delshams, Amadeu, de la Llave, Rafael, Seara, Tere M.: Geometric properties of the scattering map of a normally hyperbolic invariant manifold....
  • 8. Delshams, Amadeu, de la Llave, Rafael, Seara, Tere M.: Instability of high dimensional Hamiltonian systems: multiple resonances do not...
  • 9. Delshams, A., Gutiérrez, P.: Splitting potential and Poincaré-Melnikov method for whiskered tori in Hamiltonian systems. J. Nonlinear Sci....
  • 10. Delshams, A., Gutiérrez, P.: Homoclinic orbits to invariant tori in Hamiltonian systems. In: Christopher, K.R.T.J., Khibnik, A.I. (eds.)...
  • 11. Delshams, Amadeu, Ramírez-Ros, Rafael: Poincaré–Melnikov–Arnold method for analytic planar maps. Nonlinearity 9(1), 1–26 (1996)
  • 12. Delshams, Amadeu, Ramírez-Ros, Rafael: Melnikov potential for exact symplectic maps. Commun. Math. Phys. 190(1), 213–245 (1997)
  • 13. Delshams, Amadeu, Schaefer, Rodrigo G.: Arnold diffusion for a complete family of perturbations. Regular Chaotic Dyn. 22(1), 78–108 (2017)
  • 14. Delshams, Amadeu, Schaefer, Rodrigo G.: Arnold diffusion for a complete family of perturbations with two independent harmonics. Discret....
  • 15. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)
  • 16. Fenichel, N.: Asymptotic stability with rate conditions. Indiana Univ. Math. J. 23, 1109–1137 (1974)
  • 17. Gidea, Marian, de la Llave, Rafael: Perturbations of geodesic flows by recurrent dynamics. J. Eur. Math. Soc. (JEMS) 19(3), 905–956 (2017)
  • 18. Gidea, Marian, de la Llave, Rafael: Global Melnikov theory in Hamiltonian systems with general time-dependent perturbations. J. NonLinear...
  • 19. Gidea, M., de la Llave, R., Seara, M.T.: A general mechanism of instability in Hamiltonian systems: skipping along a normally hyperbolic...
  • 20. Gidea, Marian, de la Llave, Rafael: A general mechanism of diffusion in Hamiltonian systems: qualitative results. Commun. Pure Appl. Math....
  • 21. Guckenheimer, John, Holmes, Philip: Nonlinear oscillations, dynamical systems and bifurcations of vector fields. J. Appl. Mech. 51(4),...
  • 22. Granados, Albert, Hogan, Stephen John, Seara, Tere M: The Melnikov method and subharmonic orbits in a piecewise-smooth system. SIAM J....
  • 23. Granados, Albert, Hogan, Stephen John, Seara, T.M.: The scattering map in two coupled piecewisesmooth systems, with numerical application...
  • 24. Granados, Albert: Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators. Physica...
  • 25. Holmes, P.J., Marsden, J.E.: Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys....
  • 26. Hazeltine, Richard D., Meiss, James D.: Plasma Confinement. Courier Corporation, Chelmsford (2003)
  • 27. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lecture Notes in Math, vol. 583. Springer, Berlin (1977)
  • 28. Kyner, W.T.: Rigorous and formal stability of orbits about an oblate planet(idealized point mass motion in axisymmetric gravitational...
  • 29. Lomelı, Héctor E., Meiss, James D.: Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps....
  • 30. Lomelí, Héctor E., Meiss, James D., Ramírez-Ros, Rafael: Canonical Melnikov theory for diffeomorphisms. Nonlinearity 21(3), 485–508 (2008)
  • 31. Mel’nikov, V.K.: On the stability of a center for time-periodic perturbations. Trudy Moskov. Mat. Obšˇc. 12, 3–52 (1963)
  • 32. Pesin, Yakov B.: Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich Lectures in Advanced Mathematics. European Mathematical...
  • 33. Robinson, C.: Horseshoes for autonomous Hamiltonian systems using the Melnikov integral. Ergodic Theory Dyn. Syst. 8, 395–409 (1988)
  • 34. Roy, Nicolas: Intersections of Lagrangian submanifolds and the Melnikov 1-form. J. Geom. Phys 56, 2203–2229 (2006)
  • 35. Verhulst, Ferdinand: Nonlinear Differential Equations and Dynamical Systems. Springer, Berlin (2006)
  • 36. Wiggins, S.: Global Bifurcations and Chaos: Analytical Methods. Appl. Math. Sci., vol. 73. Springer, New York (1990)