Multibump solutions for Hamiltonian systems with fast and slow forcing

Coti Zelati, Vittorio ; Nolasco, Margherita

Bollettino della Unione matematica italiana, Série 8, 2B (1999) no. 3, pp. 585-608.

Résumé

Si dimostra l'esistenza di infinite soluzioni «multi-bump» - e conseguentemente il comportamento caotico - per una classe di sistemi Hamiltoniani del secondo ordine della forma $-\ddot{q}+q=(g_{1}(\omega t)+g_{2}(t/\omega)) V'(q)$ per $\omega$ sufficientemente piccolo. Qui $q\in \mathbb{R}^{n}$ , $g_{1}$ e $g_{2}$ sono funzioni strettamente positive e periodiche e $V$ è un potenziale superquadratico (ad esempio $V(q)=|q|^{4}$ ).

Bibliographie

[1] F. Alessio - P. Caldiroli - P. Montecchiari , Genericity of the multibump dynamics for almost periodic duffing-like systems, preprint, SISSA, Trieste (1997). | MR | Zbl

[2] A. Ambrosetti - M. Badiale - S. Cingolani , Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. | MR | Zbl

[3] A. Ambrosetti - M. Berti , Homoclinics and complex dynamics in slowly oscillating systems, Discrete Contin. Dynam. Systems (1998). | MR | Zbl

[4] S. Angenent , A variational interpretation of Melnikov's function and exponentially small separatrix splitting, Symplectic Geometry (Cambridge) (J. Mierczyński, ed.), London Math. Soc. Lecture Note, no. 192, Cambridge University Press (1993), 5-35. | MR | Zbl

[5] M. L. Bertotti - S. V. Bolotin , Homoclinic solutions of quasiperiodic Lagrangian systems, Differential Integral Equations, 8 (1995), 1733-1760. | MR | Zbl

[6] U. Bessi , Homoclinic and period-doubling bifurcations for damped systems, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 12 (1995), 1-25. | fulltext mini-dml | MR | Zbl

[7] S. Cingolani - M. Nolasco , Multi-peak periodic semiclassical states for a class of nonlinear Schroedinger equations, Proc. Roy. Soc. Edinburgh Sect. A (1998), to appear. | MR | Zbl

[8] V. Coti Zelati - I. Ekeland - E. Séré , Solutions doublement asymptotiques de systèmes Hamiltoniens convexes, C. R. Acad. Sci. Paris Sér. I Math., 310 (1990), 631-633. | MR | Zbl

[9] V. Coti Zelati - P. Montecchiari - M. Nolasco , Multibump homoclinic solutions for a class of second order, almost periodic Hamiltonian systems, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 77-99. | MR | Zbl

[10] V. Coti Zelati - P. H. Rabinowitz , Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. | MR | Zbl

[11] M. Del Pino - P. L. Felmer , Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. | MR | Zbl

[12] A. Delshams - T. M. Seara , An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys., 150 (1992), 433-463. | fulltext mini-dml | MR | Zbl

[13] B. Fiedler - J. Scheurle , Discretization of homoclinic orbits, rapid forcing and «invisible» chaos, Memoirs of the AMS, no. 570, American Mathematical Society, Providence (1996). | MR | Zbl

[14] A. Floer - A. Weinstein , Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. | MR | Zbl

[15] C. Gui , Existence of multi-bump solutions for nonlinear Schrodinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820. | MR | Zbl

[16] Y. Y. Li , On a singularly perturbed elliptic equation, preprint, Rutgers University (1997).

[17] P.-L. Lions , The concentration-compactness principle in the calculus of variations. The locally compact case, part 1., Ann. Inst. H. Poincaré. Anal. Non Linéaire, 1 (1984), 109-145. | fulltext mini-dml | MR | Zbl

[18] V. Melnikov , On the stability of the center for periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57. | MR | Zbl

[19] P. Montecchiari - M. Nolasco - S. Terracini , A global condition for periodic Duffing-like equations, Trans. Amer. Math. Soc., to appear. | MR | Zbl

[20] P. Montecchiari - M. Nolasco - S. Terracini , Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems, Calc. Var. Partial Differential Equations, 5 (1997), 423-555. | MR | Zbl

[21] Y. Oh , Existence of semiclassical bound states of nonlinear Schrödinger equations with potential in the class $(V)_\alpha$, Comm. Partial Differential Equations, 13 (1988), 1499-1519. | MR | Zbl

[22] H. Poincaré , Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris (1897-1899).

[23] P. H. Rabinowitz , Multibump solutions for an almost periodically forced singular Hamiltonian system, Electron. J. Differential Equations, 1995 (1995), no. 12, 1-21. | MR | Zbl

[24] E. Séré , Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. | MR | Zbl

[25] E. Séré , Looking for the Bernoulli shift, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 10 (1993), no. 5, 561-590. | fulltext mini-dml | MR | Zbl

[26] E. Serra - M. Tarallo - S. Terracini , On the existence of homoclinic solutions for almost periodic second order systems, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13 (1996), 783-812. | fulltext mini-dml | MR | Zbl