A global condition for periodic Duffing-like equations
HTML articles powered by AMS MathViewer
- by Piero Montecchiari, Margherita Nolasco and Susanna Terracini PDF
- Trans. Amer. Math. Soc. 351 (1999), 3713-3724 Request permission
Abstract:
We study Duffing-like equations of the type $\ddot q= q - \alpha (t)W’(q)$,with $\alpha \in C(\mathbb {R} ,\mathbb {R} )$ periodic. We prove that if the stable and unstable manifolds to the origin do not coincide, then the system exhibits positive topological entropy.References
- Melvin S. Berger, Nonlinearity and functional analysis, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis. MR 0488101
- Ugo Bessi, A variational proof of a Sitnikov-like theorem, Nonlinear Anal. 20 (1993), no. 11, 1303–1318. MR 1220837, DOI 10.1016/0362-546X(93)90133-D
- S. V. Bolotin, Existence of homoclinic motions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1983), 98–103 (Russian). MR 728558
- B. Buffoni and E. Séré, A global condition for quasi-random behavior in a class of conservative systems, Comm. Pure Appl. Math. 49 (1996), no. 3, 285–305. MR 1374173, DOI 10.1002/(SICI)1097-0312(199603)49:3<285::AID-CPA3>3.3.CO;2-
- 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
- K. Cieliebak and E. Séré, Pseudo-holomorphic curves and the shadowing lemma, Duke Math. J. (to appear).
- Vittorio Coti Zelati, Ivar Ekeland, and Éric Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288 (1990), no. 1, 133–160. MR 1070929, DOI 10.1007/BF01444526
- Vittorio Coti Zelati and Paul H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1991), no. 4, 693–727. MR 1119200, DOI 10.1090/S0894-0347-1991-1119200-3
- P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145 (English, with French summary). MR 778970, DOI 10.1016/S0294-1449(16)30428-0
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- P. Montecchiari, M. Nolasco and S. Terracini, Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems, Calc. of Var. (to appear).
- Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
- M. Nolasco, Multibump solutions for a class of time dependent second order Hamiltonian systems, Ph.D. thesis, SISSA, 1995.
- 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
- Mark Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds, London Mathematical Society Lecture Note Series, vol. 180, Cambridge University Press, Cambridge, 1993. MR 1215938, DOI 10.1017/CBO9780511752537
- Paul H. Rabinowitz, A multibump construction in a degenerate setting, Calc. Var. Partial Differential Equations 5 (1997), no. 2, 159–182. MR 1433175, DOI 10.1007/s005260050064
- Éric Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z. 209 (1992), no. 1, 27–42. MR 1143210, DOI 10.1007/BF02570817
- Éric Séré, Looking for the Bernoulli shift, Ann. Inst. H. Poincaré C Anal. Non Linéaire 10 (1993), no. 5, 561–590 (English, with English and French summaries). MR 1249107, DOI 10.1016/S0294-1449(16)30205-0
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- Stephen Wiggins, Global bifurcations and chaos, Applied Mathematical Sciences, vol. 73, Springer-Verlag, New York, 1988. Analytical methods. MR 956468, DOI 10.1007/978-1-4612-1042-9
Additional Information
- Piero Montecchiari
- Affiliation: Dipartimento di Matematica, Universitá degli studi di Trieste, Piazzale Europa 1, 34013 Trieste, Italy
- Email: montec@univ.trieste.it
- Margherita Nolasco
- Affiliation: S.I.S.S.A., via Beirut 4, 34013 Trieste, Italy
- Email: nolasco@neumann.sissa.it
- Susanna Terracini
- Affiliation: Dipartimento di Matematica del Politecnico, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
- Email: suster@ipmma1.mate.polimi.it
- Received by editor(s): July 16, 1996
- Received by editor(s) in revised form: March 31, 1997
- Published electronically: March 1, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 3713-3724
- MSC (1991): Primary 58E05, 70H35, 34C37, 58F15
- DOI: https://doi.org/10.1090/S0002-9947-99-02249-7
- MathSciNet review: 1487629