Multibump solutions for Hamiltonian systems with fast and slow forcing

Coti Zelati, Vittorio ; Nolasco, Margherita

Bollettino dell'Unione Matematica Italiana, Tome 2-A (1999), p. 585-608 / Harvested from Biblioteca Digitale Italiana di Matematica

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 $V(q)=|q|^{4}$ ).

Détail

Publié le : 1999-10-01

MR 1719562 | Zbl 0940.37008

Bibliographie

[1] Alessio, F.-Caldiroli, P.-Montecchiari, P., Genericity of the multibump dynamics for almost periodic duffing-like systems, preprint, SISSA, Trieste (1997). | MR 1719214 | Zbl 0941.34032

[2] Ambrosetti, A.-Badiale, M.-Cingolani, S., Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. | MR 1486895 | Zbl 0896.35042

[3] Ambrosetti, A.-Berti, M., Homoclinics and complex dynamics in slowly oscillating systems, Discrete Contin. Dynam. Systems (1998). | MR 1612728 | Zbl 0952.34037

[4] Angenent, S., 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 1297127 | Zbl 0810.34037

[5] Bertotti, M. L.-Bolotin, S. V., Homoclinic solutions of quasiperiodic Lagrangian systems, Differential Integral Equations, 8 (1995), 1733-1760. | MR 1347977 | Zbl 0827.34037

[6] Bessi, U., Homoclinic and period-doubling bifurcations for damped systems, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 12 (1995), 1-25. | MR 1320566 | Zbl 0836.34044

[7] Cingolani, S.-Nolasco, M., Multi-peak periodic semiclassical states for a class of nonlinear Schroedinger equations, Proc. Roy. Soc. Edinburgh Sect. A (1998), to appear. | MR 1664105 | Zbl 0922.35158

[8] Coti Zelati, V.-Ekeland, I.-Séré, E., Solutions doublement asymptotiques de systèmes Hamiltoniens convexes, C. R. Acad. Sci. Paris Sér. I Math., 310 (1990), 631-633. | MR 1065426 | Zbl 0703.70020

[9] Coti Zelati, V.-Montecchiari, P.-Nolasco, M., Multibump homoclinic solutions for a class of second order, almost periodic Hamiltonian systems, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 77-99. | MR 1433313 | Zbl 0878.34045

[10] Coti Zelati, V.-Rabinowitz, P. H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. | MR 1119200 | Zbl 0744.34045

[11] Del Pino, M.-Felmer, P. L., Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. | MR 1379196 | Zbl 0844.35032

[12] Delshams, A.-Seara, T. M., An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys., 150 (1992), 433-463. | MR 1204314 | Zbl 0765.70016

[13] Fiedler, B.-Scheurle, J., Discretization of homoclinic orbits, rapid forcing and «invisible» chaos, Memoirs of the AMS, no. 570, American Mathematical Society, Providence (1996). | MR 1342018 | Zbl 0923.34049

[14] Floer, A.-Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. | MR 867665 | Zbl 0613.35076

[15] Gui, C., Existence of multi-bump solutions for nonlinear Schrodinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820. | MR 1391524 | Zbl 0857.35116

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

[17] Lions, P.-L., 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. | MR 778970 | Zbl 0541.49009

[18] Melnikov, V., On the stability of the center for periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57. | MR 156048 | Zbl 0135.31001

[19] Montecchiari, P.-Nolasco, M.-Terracini, S., A global condition for periodic Duffing-like equations, Trans. Amer. Math. Soc., to appear. | MR 1487629 | Zbl 0926.37005

[20] Montecchiari, P.-Nolasco, M.-Terracini, S., Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems, Calc. Var. Partial Differential Equations, 5 (1997), 423-555. | MR 1473307 | Zbl 0886.58014

[21] Oh, Y., 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 970154 | Zbl 0702.35228

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

[23] Rabinowitz, P. H., Multibump solutions for an almost periodically forced singular Hamiltonian system, Electron. J. Differential Equations, 1995 (1995), no. 12, 1-21. | MR 1348521 | Zbl 0828.34034

[24] Séré, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. | MR 1143210 | Zbl 0725.58017

[25] Séré, E., Looking for the Bernoulli shift, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 10 (1993), no. 5, 561-590. | MR 1249107 | Zbl 0803.58013

[26] Serra, E.-Tarallo, M.-Terracini, S., On the existence of homoclinic solutions for almost periodic second order systems, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13 (1996), 783-812. | MR 1420498 | Zbl 0873.58032