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Melnikov transforms, Bernoulli bundles, and almost periodic perturbations


Authors: Kenneth R. Meyer and George R. Sell
Journal: Trans. Amer. Math. Soc. 314 (1989), 63-105
MSC: Primary 58F30; Secondary 34D30, 54H20, 58F13, 58F27
DOI: https://doi.org/10.1090/S0002-9947-1989-0954601-2
MathSciNet review: 954601
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Abstract: In this paper we study nonlinear time-varying perturbations of an autonomous vector field in the plane $ {R^2}$ . We assume that the unperturbed equation, i.e. the given vector field has a homoclinic orbit and we present a generalization of the Melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory. The key to our generalization is the concept of the Melnikov transform, which is a linear transformation on the space of perturbation functions.

The appropriate dynamical setting for studying these perturbation is the concept of a skew product flow. The concept of transversality we require is best understood in this context. Under conditions whereby the perturbed equation admits a transversal homoclinic trajectory, we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow. We show the dynamics near this trajectory can have the exotic behavior of the Bernoulli shift. The exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle, which we call, the Bernoulli bundle. We allow all perturbations which are bounded and uniformly continuous in time. Thus our theory includes the classical periodic perturbations studied by Melnikov, quasi periodic and almost periodic perturbations, as well as toroidal perturbations which are close to quasi periodic perturbations.


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  • [V] M. Alekseev (1976), Symbolic dynamics, "11th Math. School," Eds. Mitropolskii and Samoilenko, Kiev. MR 0464317 (57:4249)
  • [Z] Artstein (1977), Topological dynamics of ordinary differential equations and Kurzweil equations, J. Differential Equations 23, 224-243. MR 0432985 (55:5964)
  • [A] S. Besicovitch (1932), Almost periodic functions, Cambridge Univ. Press.
  • [G] D. Birkhoff (1932), Nouvelles recherches sur les systems dynamiques, Mem. Pont. Acad. Soc. Novi. Lyncaei 1, 85-216.
  • [H] Bohr (1925a), Zur Theorie der fastperiodischen Funktionen I, Acta Math. 45, 29-127. MR 1555192
  • 1. -(1925b), Zur Theorie der fastperiodischen Funktionen II, Acta Math. 46, 101-214. MR 1555201
  • 2. -(1926), Zur Theorie der fastperiodischen Funktionen III, Acta Math. 47, 237-281. MR 1555216
  • 3. -(1959), Almost periodic functions, Chelsea, New York.
  • [R] Bowen (1970), Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lectures Notes in Math., vol. 470, Springer-Verlag, New York. MR 2423393 (2009d:37038)
  • [M] L. Cartwright and J. E. Littlewood (1945), On non-linear differential equations of second order. I. The equation $ \ddot y - k(1 - {y^2})\dot y + y = b\lambda k\;\cos (\lambda t + a),k$ large, J. London Math. Soc. 20, 180-189. MR 0016789 (8:68g)
  • [S] N. Chow and J. K. Hale (1982), Methods of bifurcation theory, Springer-Verlag, New York. MR 660633 (84e:58019)
  • [S] N. Chow, J. K. Hale and J. Mallet-Paret (1980), An example of bifurcation to homoclinic orbits, J. Differential Equations 37, 351-373. MR 589997 (81m:58056)
  • [C] C. Conley (1978), Isolated invariant sets and the morse index, CBMS Regional Conf. Ser. Math., no. 38, Amer. Math. Soc, Providence, R. I. MR 511133 (80c:58009)
  • [W] A. Coppel (1965), Stability and asymptotic behavior of differential equations, Heath, Boston, Mass. MR 0190463 (32:7875)
  • 4. -(1978), Dichotomies in stability theory, Lecture Notes in Math., vol. 206, Springer-Verlag, New York. MR 0481196 (58:1332)
  • [C] Corduneanu (1968), Almost periodic functions, Interscience, New York. MR 0481915 (58:2006)
  • [I] P. Cornfeld, S. V. Fomin and Ya. G. Sinai (1982), Ergodic theory, Springer-Verlag, New York. MR 832433 (87f:28019)
  • [R] Devaney (1986), An introduction to chaotic dynamical systems, Benjamin Cummings, Menlo Park, California. MR 811850 (87e:58142)
  • [N] Ercolani, M. G. Forest and D. W. McLaughlin (1987), Homoclinic orbits for the periodic sine-Gordon equation, preprint.
  • [J] Favard (1933), Leçons sur les fonctions presque périodiques, Gauthier-Villars, Paris.
  • [A] M. Fink (1974), Almost periodic differential equations, Lecture Notes in Math., vol. 377, Springer-Verlag, New York. MR 0460799 (57:792)
  • [J] Guckenheimer and P. Holmes (1983), Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York. MR 709768 (85f:58002)
  • [J] K. Hale (1969), Ordinary differential equations, Wiley-Interscience, New York. MR 0419901 (54:7918)
  • [J] G. Hocking and G. S. Young (1961), Topology, Addison-Wesley, New York. MR 0125557 (23:A2857)
  • [P] Holmes (1980), Averaging and chaotic motion in forced oscillations, SIAM Appl. Math. 38, 68-80. MR 559081 (80m:34044)
  • [G] Ikegami (1969), On classification of dynamical systems with cross sections, Osaka Math. J. 6, 419-433. MR 0266224 (42:1131)
  • [B] M. Levitan and V. V. Zhikov (1982), Almost periodic functions and differential equations, Cambridge Univ. Press., Cambridge. MR 690064 (84g:34004)
  • [L] Markus and K. R. Meyer (1980), Periodic orbits and solenoids in generic Hamiltonian systems, Amer. J. Math. 102, 25-92. MR 556887 (81g:58013)
  • [J] E. Marsden (1984), Chaos in dynamical systems by the Poincaré-Melnikov-Arnold method, Proc. ARO Workshop on Dynamics, SIAM. MR 831471 (87i:58121)
  • [V] K. Melnikov (1963), On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12, 1-57. MR 0156048 (27:5981)
  • [K] R. Meyer and G. R. Sell (1986), Homoclinic orbits and Bernoulli bundles in almost periodic systems, IMA Preprint No. 285, November, 1986, Oscillations, Bifurcations and Chaos, CBMS Conference Proceedings, pp. 527-544. MR 909934 (89a:34050)
  • 5. -(1987), An analytic proof of the shadowing lemma, Funkcial. Ekvac. 30, 127-133. MR 915267 (89b:58169)
  • 6. -(1989), A model for describing chaos in the perturbed sine-Gordon equation.
  • [R] K. Miller (1965), Almost periodic differential equations as dynamical systems with applications to the existence of a.p. solutions, J. Differential Equations 1, 337-345. MR 0185221 (32:2690)
  • [R] K. Miller and G. R. Sell (1970), Volterra integral equations and topological dynamics, Mem. Amer. Math. Soc. No. 102. MR 0288381 (44:5579)
  • [M] Morse (1921), A one-to-one representation of geodesies on a surface of negative curvature, Amer. J. Math. 43, 33-51. MR 1506428
  • [V] V. Nemytskii and V. V. Stepanov (1960), Qualitative theory of differential equations, Princeton Univ., Princeton, N. J. MR 0121520 (22:12258)
  • [D] A. Neumann (1976), Dynamical systems with cross sections, Proc. Amer. Math. Soc. 58, 339-344. MR 0407903 (53:11673)
  • [J] Palis and W. de Meló (1980), Geometric theory of dynamical systems, Springer-Verlag, New York.
  • [K] J. Palmer (1984), Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55, 225-156. MR 764125 (86d:58088)
  • [H] Poincaré (1892), Les méthodes nouvelles de la mécanique céleste III, Gauthier-Villars, Paris.
  • [H] G. Poinkhoff (1973), An analytic closing lemma, "Proc. Midwest Dyn. Sys. Sem.," Northwestern Univ. Press, Evanston, Ill., pp. 128-256.
  • [L] S. Pontryagin (1966), Topological groups, 2nd ed., Gordon and Breach, New York. MR 0201557 (34:1439)
  • [R] J. Sacker and G. R. Sell (1974), Existence of dichotomies and invariant splitting for linear differential systems I, J. Differential Equations 15, 429-458. MR 0341458 (49:6209)
  • 7. -(1976), Existence of dichotomies and invariant splitting for linear differential systems II, J. Differential Equations 22, 478-496. MR 0440620 (55:13494)
  • 8. -(1977), Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc. No. 190. MR 0448325 (56:6632)
  • [J] Scheurle (1986), Chaotic solutions of systems with almost periodic forcing, preprint. MR 831922 (87k:58234)
  • [S] Schwabik (1985), Generalized differential equations, Czechoslovakian Akad. Sci., Prague.
  • [G] R. Sell (1967), Nonautonomous differential equations and topological dynamics I, II, Trans. Amer. Math. Soc. 127, 241-262 and 263-283.
  • 9. -(1971), Topological dynamics and ordinary differential equations, Van Nostrand, New York. MR 0442908 (56:1283)
  • 10. -(1978), The structure of a flow in the vicinity of an almost periodic motion, J. Differential Equations 27(3), 359-93. MR 0492608 (58:11704)
  • [M] Shub (1987), Global stability of dynamical systems, Springer-Verlag, New York. MR 869255 (87m:58086)
  • 11. Ya. G. Sinai (1973), Introduction to ergodic theory, English transl., Princeton Univ. Press, Princeton, N. J. MR 0584788 (58:28437)
  • [S] Smale (1963), Diffeomorphisms with infinitely many periodic points, Differential and Combinatorial Topology, Ed., S. Chern, Princeton Univ. Press, pp. 63-80. MR 0182020 (31:6244)
  • [S] Wiggens (1986a), A generalization of the method of Melnikov for detecting chaotic invariant sets, preprint.
  • 12. -(1986b), The orbit structure in the neighborhood of a transverse homoclinic torus, preprint.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0954601-2
Keywords: Almost periodic, Bernoulli shift, exponential dichotomy, homoclinic orbit, Melnikov method, shadowing lemma
Article copyright: © Copyright 1989 American Mathematical Society

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