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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Mel′nikov transforms, Bernoulli bundles, and almost periodic perturbations
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by Kenneth R. Meyer and George R. Sell PDF
Trans. Amer. Math. Soc. 314 (1989), 63-105 Request permission

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.
References
  • V. M. Alekseev, Symbolic dynamics, Eleventh Mathematical School (Summer School, Kolomyya, 1973) Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1976, pp. 5–210 (Russian). MR 0464317
  • Zvi Artstein, Topological dynamics of ordinary differential equations and Kurzweil equations, J. Differential Equations 23 (1977), no. 2, 224–243. MR 432985, DOI 10.1016/0022-0396(77)90128-0
    • S. Besicovitch (1932), Almost periodic functions, Cambridge Univ. Press. D. Birkhoff (1932), Nouvelles recherches sur les systems dynamiques, Mem. Pont. Acad. Soc. Novi. Lyncaei 1, 85-216.
  • Harald Bohr, Zur theorie der fast periodischen funktionen, Acta Math. 45 (1925), no. 1, 29–127 (German). I. Eine verallgemeinerung der theorie der fourierreihen. MR 1555192, DOI 10.1007/BF02395468
  • Harald Bohr, Zur Theorie der Fastperiodischen Funktionen, Acta Math. 46 (1925), no. 1-2, 101–214 (German). II. Zusammenhang der fastperiodischen Funktionen mit Funktionen von unendlich vielen Variabeln; gleichmässige Approximation durch trigonometrische Summen. MR 1555201, DOI 10.1007/BF02543859
  • Harald Bohr, Zur Theorie der fastperiodischen Funktionen, Acta Math. 47 (1926), no. 3, 237–281 (German). III. Dirichletentwicklung analytischer Funktionen. MR 1555216, DOI 10.1007/BF02543846
    • —(1959), Almost periodic functions, Chelsea, New York.
  • Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 2008. With a preface by David Ruelle; Edited by Jean-René Chazottes. MR 2423393, DOI 10.1007/978-3-540-77695-6
  • M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order. I. The equation $\ddot y-k(1-y^2)y+y=b\lambda k\;\textrm {cos} (\lambda t+a), k$ large, J. London Math. Soc. 20 (1945), 180–189. MR 16789, DOI 10.1112/jlms/s1-20.3.180
  • Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633, DOI 10.1007/978-1-4613-8159-4
  • Shui Nee Chow, Jack K. Hale, and John Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations 37 (1980), no. 3, 351–373. MR 589997, DOI 10.1016/0022-0396(80)90104-7
  • Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133, DOI 10.1090/cbms/038
  • W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Company, Boston, Mass., 1965. MR 0190463
  • W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR 0481196
  • C. Corduneanu, Almost periodic functions, Interscience Tracts in Pure and Applied Mathematics, No. 22, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. With the collaboration of N. Gheorghiu and V. Barbu; Translated from the Romanian by Gitta Bernstein and Eugene Tomer. MR 0481915
  • I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
  • Robert L. Devaney, An introduction to chaotic dynamical systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986. MR 811850
    • Ercolani, M. G. Forest and D. W. McLaughlin (1987), Homoclinic orbits for the periodic sine-Gordon equation, preprint. Favard (1933), Leçons sur les fonctions presque périodiques, Gauthier-Villars, Paris.
  • A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799, DOI 10.1007/BFb0070324
  • John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
  • Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. MR 0419901
  • John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
  • Philip J. Holmes, Averaging and chaotic motions in forced oscillations, SIAM J. Appl. Math. 38 (1980), no. 1, 65–80. MR 559081, DOI 10.1137/0138005
  • Gikô Ikegami, On classification of dynamical systems with cross-sections, Osaka Math. J. 6 (1969), 419–433. MR 266224
  • B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR 690064
  • L. Markus and Kenneth R. Meyer, Periodic orbits and solenoids in generic Hamiltonian dynamical systems, Amer. J. Math. 102 (1980), no. 1, 25–92. MR 556887, DOI 10.2307/2374171
  • Jerrold E. Marsden, Chaos in dynamical systems by the Poincaré-Mel′nikov-Arnol′d method, Chaos in nonlinear dynamical systems (Research Triangle Park, N.C., 1984) SIAM, Philadelphia, PA, 1984, pp. 19–31. MR 831471
  • V. K. Mel′nikov, On the stability of a center for time-periodic perturbations, Trudy Moskov. Mat. Obšč. 12 (1963), 3–52 (Russian). MR 0156048
  • Kenneth R. Meyer and George R. Sell, Homoclinic orbits and Bernoulli bundles in almost periodic systems, Oscillations, bifurcation and chaos (Toronto, Ont., 1986) CMS Conf. Proc., vol. 8, Amer. Math. Soc., Providence, RI, 1987, pp. 527–544. MR 909934
  • K. R. Meyer and George R. Sell, An analytic proof of the shadowing lemma, Funkcial. Ekvac. 30 (1987), no. 1, 127–133. MR 915267
    • —(1989), A model for describing chaos in the perturbed sine-Gordon equation.
  • R. K. Miller, Almost periodic differential equations as dynamical systems with applications to the existence of A.P. solutions, J. Differential Equations 1 (1965), 337–345. MR 185221, DOI 10.1016/0022-0396(65)90012-4
  • Richard K. Miller and George R. Sell, Volterra integral equations and topological dynamics, Memoirs of the American Mathematical Society, No. 102, American Mathematical Society, Providence, R.I., 1970. MR 0288381
  • Harold Marston Morse, A One-to-One Representation of Geodesics on a Surface of Negative Curvature, Amer. J. Math. 43 (1921), no. 1, 33–51. MR 1506428, DOI 10.2307/2370306
  • V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR 0121520
  • Dean A. Neumann, Dynamical systems with cross-sections, Proc. Amer. Math. Soc. 56 (1976), 339–344. MR 407903, DOI 10.1090/S0002-9939-1976-0407903-9
    • Palis and W. de Meló (1980), Geometric theory of dynamical systems, Springer-Verlag, New York.
  • Kenneth J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), no. 2, 225–256. MR 764125, DOI 10.1016/0022-0396(84)90082-2
    • Poincaré (1892), Les méthodes nouvelles de la mécanique céleste III, Gauthier-Villars, Paris. G. Poinkhoff (1973), An analytic closing lemma, "Proc. Midwest Dyn. Sys. Sem.," Northwestern Univ. Press, Evanston, Ill., pp. 128-256.
  • L. S. Pontryagin, Topological groups, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR 0201557
  • Robert J. Sacker and George R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations 15 (1974), 429–458. MR 341458, DOI 10.1016/0022-0396(74)90067-9
  • Robert J. Sacker and George R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. II, J. Differential Equations 22 (1976), no. 2, 478–496. MR 440620, DOI 10.1016/0022-0396(76)90042-5
  • Robert J. Sacker and George R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc. 11 (1977), no. 190, iv+67. MR 448325, DOI 10.1090/memo/0190
  • Jürgen Scheurle, Chaotic solutions of systems with almost periodic forcing, Z. Angew. Math. Phys. 37 (1986), no. 1, 12–26. MR 831922, DOI 10.1007/BF00955515
    • Schwabik (1985), Generalized differential equations, Czechoslovakian Akad. Sci., Prague. R. Sell (1967), Nonautonomous differential equations and topological dynamics I, II, Trans. Amer. Math. Soc. 127, 241-262 and 263-283.
  • George R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold Mathematical Studies, No. 33, Van Nostrand Reinhold Co., London, 1971. MR 0442908
  • George R. Sell, The structure of a flow in the vicinity of an almost periodic motion, J. Differential Equations 27 (1978), no. 3, 359–393. MR 492608, DOI 10.1016/0022-0396(78)90058-X
  • Michael Shub, Global stability of dynamical systems, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR 869255, DOI 10.1007/978-1-4757-1947-5
  • Ya. G. Sinai, Introduction to ergodic theory, Mathematical Notes, vol. 18, Princeton University Press, Princeton, N.J., 1976. Translated by V. Scheffer. MR 0584788
  • Stephen Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR 0182020
    • Wiggens (1986a), A generalization of the method of Melnikov for detecting chaotic invariant sets, preprint. —(1986b), The orbit structure in the neighborhood of a transverse homoclinic torus, preprint.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • 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