Advertisement

Regular and Chaotic Dynamics

, Volume 24, Issue 4, pp 392–417 | Cite as

On Transversal Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field: the Newton – Kantorovich Approach

  • Alexey V. IvanovEmail author
Article

Abstract

We consider a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q,t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t → ±∞ and vanishes at a unique point t0 ∈ ℝ. Let X+, X denote the sets of isolated critical points of V(x) at which U(x,t) as a function of x attains its maximum for any fixed t > t0 and t < t0, respectively. Under nondegeneracy conditions on points of X± we apply the Newton – Kantorovich type method to study the existence of transversal doubly asymptotic trajectories connecting X and X+. Conditions on the Riemannian manifold and the potential which guarantee the existence of such orbits are presented. Such connecting trajectories are obtained by continuation of geodesies defined in a vicinity of the point t0 to the whole real line.

Keywords

connecting orbits homoclinics heteroclinics nonautonomous Lagrangian system Newton – Kantorovich method 

MSC2010 numbers

37J45 37C29 58K45 65P10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding

This research was supported by RFBR grant (project No. 17-01-00668/19).

References

  1. 1.
    Angenent, S., A Variational Interpretation of Melnikov’s Function and Exponentially Small Separatrix Splitting, in Symplectic Geometry, D. Salamon (Ed.), London Math. Soc. Lecture Note Ser., vol.192, Cambridge: Cambridge Univ. Press, 1993, pp. 5–35.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Atkin, C. J., The Hopf-Rinow Theorem Is False in Infinite Dimensions, Bull. London Math. Soc, 1975. vol. 7, no. 3, pp. 261–266.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertotti, M.L. and Bolotin, S.V., Doubly Asymptotic Trajectories of Lagrangian Systems in Homogeneous Force Fields, Ann. Mat. Pura Appl (4)., 1998, vol. 174, pp. 253–275.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bessi, U., An Approach to Arnold’s Diffusion through the Calculus of Variations, Nonlinear Anal, 1996. vol.26, no. 6, pp. 1115–1135.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Birkhoff, G. D., Dynamical Systems, Providence, R.I.: AMS, 1966.zbMATHGoogle Scholar
  6. 6.
    Bolotin, S.V. and Kozlov, V. V., Asymptotic Solutions of the Equations of Dynamics, Mosc. Univ. Mech. Bull, 1980, vol. 35, nos. 3-4, pp. 82–88. see also: Vestn. Most Univ. Ser. 1. Mat. Mekh., 1980. no. 4, pp. 84-89. 102.zbMATHGoogle Scholar
  7. 7.
    Borisov, A.V., Kozlov, V. V., and Mamaev, I. S., Asymptotic Stability and Associated Problems of Dynamics of Falling Rigid Body, Regal. Chaotic Dyn., 2007, vol. 12, no. 5, pp. 531–565.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coomes, B. A., Kocak, H., and Palmer, K. J., Homoclinic Shadowing, J. Dynam. Differential Equations, 2005, vol. 17, no. 1, pp. 175–215.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Coti Zelati, V. and Rabinowitz, P. H., Heteroclinic Solutions between Stationary Points at Different Energy Levels, Topol. Methods Nonlinear Anal., 2001, vol. 17, no. 1, pp. 1–21.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ekeland, I., The Hopf-Rinow Theorem in Infinite Dimension, J. Differential Geom., 1978, vol. 13, no. 2. pp. 287–301.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eliasson, H. I., Condition (C) and Geodesics on Sobolev Manifolds, Bull. Amer. Math. Soc, 1971, vol. 77, pp.1002–1005.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gelfreich, V. G. and Lazutkin, V. F., Splitting of Separatrices: Perturbation Theory and Exponential Smallness, Russian Math. Surveys, 2001, vol.56, no. 3, pp. 499–558. see also: Uspekhi Mat. Nauk, 2001, vol.56, no. 3(339), pp. 79-142.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Giannoni, F., On the Existence of Homoclinic Orbits on Riemannian Manifolds, Ergodic Theory Dynam. Systems, 1994, vol. 14, no. 1, pp. 103–127.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Giannoni, F. and Rabinowitz, P. H., On the Multiplicity of Homoclinic Orbits on Riemannian Manifolds for a Class of Second Order Hamiltonian Systems, NoDEA Nonlinear Differential Equations Appl, 1994, vol. 1, no. 1, pp. 1–46.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grossman, N., Hilbert Manifolds without Epiconjugate Points, Proc. Araer. Math. Soc, 1965, vol. 16. pp.1365–1371.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fenichel, N., Geometric Singular Perturbation Theory for Ordinary Differential Equations, J. Differential Equations, 1979, vol.31, no. 1, pp. 53–98.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ferreira, O. P. and Svaiter, B. F., Kantorovich’s Theorem on Newton’s Method in Riemannian Manifolds. J. Complexity, 2002, vol. 18, no. 1, pp. 304–329.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ivanov, A. V., Connecting Orbits of Lagrangian Systems in a Nonstationary Force Field, Regul. Chaotic Dyn., 2016, vol.21, no. 5, pp. 510–522.Google Scholar
  19. 19.
    Ivanov, A. V., Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points. Regul. Chaotic Dyn., 2017, vol.22, no. 5, pp. 479–501.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Izydorek, M. and Janczewska, J., Heteroclinic Solutions for a Class of the Second Order Hamiltonian Systems, J. Differential Equations, 2007, vol. 238, no. 2, pp. 381–393.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kantorovich, L. V., Further Applications of Newton’s Method, Vestn. Leningr. Univ., 1957, vol.2, no. 2, pp. 68–103.(Russian).Google Scholar
  22. 22.
    Kozlov, V. V., On Falling of a Heavy Rigid Body in an Ideal Fluid, Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 1989, no. 5, pp. 10–17.(Russian).Google Scholar
  23. 23.
    Kozlov, V. V., On the Stability of Equilibrium Positions in Non-Stationary Force Fields, J. Appl. Math. Meek, 1991, vol.55, no. 1, pp. 14–19. see also: Prikl. Mai. Mekh., 1991, vol.55, no. 1, pp. 12-19.Google Scholar
  24. 24.
    Lang, S., Fundamentals of Differential Geometry, 3rd ed., Grad. Texts in Math., vol. 191, New York: Springer, 1999.Google Scholar
  25. 25.
    Maeda, Y., Rosenberg, S., and Torres-Ardila, F., The Geometry of Loop Spaces: 1. Hs-Riemannian Metrics, Internal J. Math., 2015, vol.26, no.4, 15400029, 26pp.Google Scholar
  26. 26.
    Melnikov, V. K., On the Stability of the Center for Time Periodic Perturbations, Trans. Moscow Math. Soc, 1963, vol. 12, pp. 1–57. see also: Tr. Most Mat. Obs., 1963, vol. 12, pp. 3-52.MathSciNetGoogle Scholar
  27. 27.
    Palais, R. S., Morse Theory on Hilbert Manifolds, Topology, 1963, vol. 2, pp. 299–340.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Palmer, K., Shadowing in Dynamical Systems: Theory and Applications, Math. Appl., vol.501, Boston, Mass.: Kluwer, 2000.Google Scholar
  29. 29.
    Poincare, H., Les methodes nouvelles de la mecanique celeste: In 3 Vols., Paris: Gauthier-Villars, 1892, 1893, 1899. (New York: Dover, 1957; reprint.)Google Scholar
  30. 30.
    Rabinowitz, P. H., Periodic and Heteroclinic Orbits for a Periodic Hamiltonian System, Ann. Inst. H. Poincare Anal. Non Lineaire, 1989, vol. 6, no. 5, pp. 331–346.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Serre, J.-P., Homologie singuliere des espaces fibres: Applications, Ann. of Math. (2), 1951, vol.54, pp.425–505.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Smale, S., Diffeomorphisms with Many Periodic Points, in Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, S.S. Cairns (Ed.), Princeton, N.J.: Princeton Univ. Press. 1965, pp. 63–80.CrossRefGoogle Scholar
  33. 33.
    Treschev, D.V., An Averaging Method for Hamiltonian Systems, Exponentially Close to Integrable Ones, Chaos, 1996, vol.6, no. 1, pp. 6–14.Google Scholar
  34. 34.
    Wiggins, S., Global Bifurcations and Chaos. Analytical Methods, Appl. Math. Sci., vol.73, New York: Springer, 1988.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations