On global non-degeneracy conditions for chaotic behavior for a class of dynamical systems
Montecchiari, Piero1 ; Rabinowitz, Paul H.2
1 Dipartimento di Ingegneria Civile, Edile e Architettura, Università Politecnica delle Marche, Via brecce bianche, Ancona, I-60131, Italy
2 Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 627-653.
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For about 25 years, global methods from the calculus of variations have been used to establish the existence of chaotic behavior for some classes of dynamical systems. Like the analytical approaches that were used earlier, these methods require nondegeneracy conditions, but of a weaker nature than their predecessors. Our goal here is study such a nondegeneracy condition that has proved useful in several contexts including some involving partial differential equations, and to show this condition has an equivalent formulation involving stable and unstable manifolds.

DOI : 10.1016/j.anihpc.2018.08.002
Classification : 35J50, 35J47, 35J57, 34C37
Mots-clés : Chaotic behavior, Non-degeneracy conditions, Calculus of variations, Hamiltonian systems, Heteroclinic, Homoclinic
Montecchiari, Piero 1 ; Rabinowitz, Paul H. 2

1 Dipartimento di Ingegneria Civile, Edile e Architettura, Università Politecnica delle Marche, Via brecce bianche, Ancona, I-60131, Italy
2 Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA 
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Montecchiari, Piero; Rabinowitz, Paul H. On global non-degeneracy conditions for chaotic behavior for a class of dynamical systems. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 3, pp. 627-653. doi : 10.1016/j.anihpc.2018.08.002. http://www.numdam.org/articles/10.1016/j.anihpc.2018.08.002/

[1] Bessi, U. A variational proof of a Sitnikov-like theorem, Nonlinear Anal., Volume 20 (1993), pp. 1303–1318 | DOI | MR | Zbl

[2] Byeon, J.; Montecchiari, P.; Rabinowitz, P.H. A double well potential system, Anal. PDE, Volume 9 (2016), pp. 1737–1772 | DOI | MR | Zbl

[3] Buffoni, B.; Séré, E. A global condition for quasi-random behaviour in a class of conservative systems, Commun. Pure Appl. Math., Volume 49 (1996), pp. 285–305 | DOI | MR | Zbl

[4] Cieliebak, K.; Séré, E. Pseudoholomorphic curves and the shadowing lemma, Duke Math. J., Volume 99 (1999), pp. 41–73 | DOI | MR | Zbl

[5] Coti Zelati, V.; Rabinowitz, P.H. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Am. Math. Soc., Volume 4 (1991), pp. 693–727 | DOI | MR | Zbl

[6] Montecchiari, P.; Rabinowitz, P.H. On the existence of multi-transition solutions for a class of elliptic systems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016), pp. 199–219 | Numdam | MR | Zbl

[7] Montecchiari, P.; Rabinowitz, P.H. Solutions of mountain pass type for double well potential systems, Calc. Var. Partial Differ. Equ., Volume 57 (2018), pp. 114 | DOI | MR | Zbl

[8] Montecchiari, P.; Nolasco, M.; Terracini, S. Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems, Calc. Var. Partial Differ., Volume 5 (1997) no. 6, pp. 523–555 | MR | Zbl

[9] Montecchiari, P.; Nolasco, M.; Terracini, S. A global condition for periodic Duffing-like equations, Trans. Am. Math. Soc., Volume 351 (1999) no. 9, pp. 3713–3724 | DOI | MR | Zbl

[10] Kirchgraber, U.; Stoffer, D. Chaotic behaviour in simple dynamical systems, SIAM Rev., Volume 32 (1990) no. 3, pp. 424–452 | DOI | MR | Zbl

[11] Pohozaev, S.I. The set of critical values of a functional, Math. USSR Sb., Volume 4 (1968), pp. 93–98 | DOI | Zbl

[12] Poincaré, H. Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris, 1899 | JFM

[13] Rabinowitz, P.H. Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Var. Partial Differ. Equ., Volume 1 (1993), pp. 1–36 | DOI | MR | Zbl

[14] Rabinowitz, P.H. A multibump construction in a degenerate setting, Calc. Var. Partial Differ. Equ., Volume 5 (1997), pp. 15–182 | DOI | MR | Zbl

[15] Rabinowitz, P.H. Connecting orbits for a reversible Hamiltonian system, Ergod. Theory Dyn. Syst., Volume 20 (2000), pp. 1767–1784 | DOI | MR | Zbl

[16] Séré, E. Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., Volume 209 (1992), pp. 27–42 | DOI | MR | Zbl

[17] Séré, E. Looking for the Bernoulli shift, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 10 (1993), pp. 561–590 | DOI | Numdam | MR | Zbl

[18] Whyburn, G.T. Topological Analysis, Princeton University Press, Princeton, 1964 | DOI | MR | Zbl

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