Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T04:08:46.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems

Published online by Cambridge University Press:  26 February 2014

LIVIA CORSI  and
GUIDO GENTILE
LIVIA CORSI
Affiliation:
Dipartimento di Matematica, Università di Roma Tre, I-00146 Roma, Italy email [email protected], [email protected]
GUIDO GENTILE
Affiliation:
Dipartimento di Matematica, Università di Roma Tre, I-00146 Roma, Italy email [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeros of a suitable function, called the Melnikov function—by analogy with the periodic case. We show that, if the Melnikov function has a zero of odd order and under some further condition on the sign of the perturbation parameter, then there exists at least one resonant solution which continues an unperturbed solution. If the Melnikov function is identically zero then one can push perturbation theory up to the order where a counterpart of Melnikov function appears and does not vanish identically: if such a function has a zero of odd order and a suitable positiveness condition is met, again the same persistence result is obtained. If the system is Hamiltonian, then the procedure can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the result follows that at least one resonant quasi-periodic solution always exists with no assumption on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 4 , June 2015 , pp. 1079 - 1140
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andronov, A. A., Leontovich, E. A., Gordon, I. I. and Maĭer, A. G.. Theory of Bifurcations of Dynamic Systems on a Plane. Halsted Press, Israel Program for Scientific Translations, Jerusalem, 1973.Google Scholar
Arnold, V. I.. Instability of dynamical systems with several degrees of freedom. Soviet Math. Dokl. 5 (1964), 581–585.Google Scholar
Ambrosetti, A. and Badiale, M.. Homoclinics: Poincaré–Melnikov type results via a variational approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(2) (1998), 233–252.CrossRefGoogle Scholar
Ambrosetti, A., Coti Zelati, V. and Ekeland, I.. Symmetry breaking in Hamiltonian systems. J. Differential Equations 67(2) (1987), 165–184.CrossRefGoogle Scholar
Bartuccelli, M. V., Deane, J. H. B. and Gentile, G.. Bifurcation phenomena and attractive periodic solutions in the saturating inductor circuit. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463(2085) (2007), 2351–2369.Google Scholar
Bartuccelli, M. V. and Gentile, G.. Lindstedt series for perturbations of isochronous systems: a review of the general theory. Rev. Math. Phys. 14(2) (2002), 121–171.CrossRefGoogle Scholar
Belhaq, M. and Houssni, M.. Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Nonlinear Dynam. 18(1) (1999), 1–24.CrossRefGoogle Scholar
Berti, M. and Bolle, Ph.. A functional analysis approach to Arnold diffusion. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(4) (2002), 395–450.CrossRefGoogle Scholar
Berti, M. and Bolle, P.. Homoclinics and chaotic behaviour for perturbed second order systems. Ann. Mat. Pura Appl. (4) 176 (1999), 323–378.CrossRefGoogle Scholar
Berti, M., Biasco, L. and Bolle, Ph.. Drift in phase space: a new variational mechanism with optimal diffusion time. J. Math. Pures Appl. 82(6) (2003), 613–664.CrossRefGoogle Scholar
Bessi, U.. An approach to Arnold’s diffusion through the calculus of variations. Nonlinear Anal. 26(6) (1996), 1115–1135.CrossRefGoogle Scholar
Bricmont, J., Gawędzki, K. and Kupiainen, A.. KAM theorem and quantum field theory. Comm. Math. Phys. 201(3) (1999), 699–727.CrossRefGoogle Scholar
Broer, H., Hanssmann, H., Jorba, À., Villanueva, J. and Wagener, F.. Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach. Nonlinearity 16(5) (2003), 1751–1791.CrossRefGoogle Scholar
Bryuno, A. D.. Analytic form of differential equations. I. Trudy Moskov. Mat. Obšč 25 (1971), 119–262 Engl. transl. Trans. Moscow Math. Soc. 25 (1971), 131–288.Google Scholar
Bryuno, A. D.. Analytic form of differential equations. II. Trudy Moskov. Mat. Obšč 26 (1972), 199–239 Engl. transl. Trans. Moscow Math. Soc. 26 (1972), 199–239.Google Scholar
Cheng, Ch.-Q.. Birkhoff–Kolmogorov–Arnold–Moser tori in convex Hamiltonian systems. Comm. Math. Phys. 177(3) (1996), 529–559.CrossRefGoogle Scholar
Coppel, W. A.. Dichotomies in Stability Theory (Lecture Notes in Mathematics, 629). Springer, Berlin, 1978.CrossRefGoogle Scholar
Coti Zelati, V., Ekeland, I. and Séré, É. A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288(1) (1990), 133–160.CrossRefGoogle Scholar
Corsi, L. and Gentile, G.. Melnikov theory to all orders and Puiseux series for subharmonic solutions. J. Math. Phys. 49(11) (2009), 112701 29 pp.CrossRefGoogle Scholar
Corsi, L. and Gentile, G.. Oscillation synchronisation under arbitrary quasi-periodic forcing. Comm. Math. Phys. 316(2) (2012), 489–529.CrossRefGoogle Scholar
Corsi, L., Gentile, G. and Procesi, M.. KAM theory in configuration space and cancellations in the Lindstedt series. Comm. Math. Phys. 302(2) (2011), 359–402.CrossRefGoogle Scholar
Delshams, A., Gelfreich, V., Jorba, À. and Seara, T. M.. Exponentially small splitting of separatrices under fast quasiperiodic forcing. Comm. Math. Phys. 189(1) (1997), 35–71.CrossRefGoogle Scholar
Delshams, A., de la Llave, R. and Seara, T. M.. A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. Mem. Amer. Math. Soc. 179(844) (2006), viii+141 pp.Google Scholar
De Simone, E.. A renormalization proof of the KAM theorem for non-analytic perturbations. Rev. Math. Phys. 19(6) (2007), 639–675.CrossRefGoogle Scholar
De Simone, E. and Kupiainen, A.. The KAM theorem and renormalization group. Ergod. Th. & Dynam. Sys. 29(2) (2009), 419–431.CrossRefGoogle Scholar
Fenichel, N.. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21 (1971), 193–226.CrossRefGoogle Scholar
Franz, A.. Hausdorff dimension estimates for non-injective maps using the cardinality of the pre-image sets. Nonlinearity 13(5) (2000), 1425–1438.CrossRefGoogle Scholar
Gallavotti, G.. Arnold’s diffusion in isochronous systems. Math. Phys. Anal. Geom. 1(4) (1999), 295–312.CrossRefGoogle Scholar
Gallavotti, G. and Gentile, G.. Hyperbolic low-dimensional invariant tori and summation of divergent series. Comm. Math. Phys. 227(3) (2002), 421–460.CrossRefGoogle Scholar
Gallavotti, G., Gentile, G. and Giuliani, A.. Fractional Lindstedt series. J. Math. Phys. 47(1) (2006), 012702; 33 pp.CrossRefGoogle Scholar
Gallavotti, G., Gentile, G. and Mastropietro, V.. Separatrix splitting for systems with three time scales. Comm. Math. Phys. 202 (1999), 197–236.CrossRefGoogle Scholar
Gallavotti, G., Gentile, G. and Mastropietro, V.. Melnikov’s approximation dominance. Some examples. Rev. Math. Phys. 11(4) (1999), 451–461.CrossRefGoogle Scholar
Gallavotti, G., Gentile, G. and Mastropietro, V.. Hamilton–Jacobi equation, heteroclinic chains and Arnol’d diffusion in three time scales systems. Nonlinearity 13 (2000), 323–340.CrossRefGoogle Scholar
Gentile, G.. Resummation of perturbation series and reducibility for Bryuno skew-product flows. J. Stat. Phys. 125(2) (2006), 321–361.CrossRefGoogle Scholar
Gentile, G.. Degenerate lower-dimensional tori under the Bryuno condition. Ergod. Th. & Dynam. Sys. 27(2) (2007), 427–457.CrossRefGoogle Scholar
Gentile, G.. Quasi-periodic motions in strongly dissipative forced systems. Ergod. Th. & Dynam. Sys. 30(5) (2010), 1457–1469.CrossRefGoogle Scholar
Gentile, G., Bartuccelli, M. V. and Deane, J. H. B.. Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies. Rev. Math. Phys. 19(3) (2007), 307–348.CrossRefGoogle Scholar
Guckenheimer, J. and Holmes, Ph.. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences, 42). Springer, New York, 1990.Google Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
Khanin, K., Lopes Dias, J. and Marklof, J.. Multidimensional continued fractions, dynamical renormalization and KAM theory. Comm. Math. Phys. 270(1) (2007), 197–231.CrossRefGoogle Scholar
Koch, H.. A renormalization group for Hamiltonians, with applications to KAM tori. Ergod. Th. & Dynam. Sys. 19 (1999), 475–521.CrossRefGoogle Scholar
Koch, H. and Kocić, S.. A renormalization approach to lower-dimensional tori with Brjuno frequency vectors. J. Differential Equations 249(8) (2010), 1986–2004.CrossRefGoogle Scholar
Kevorkian, J. and Cole, J. D.. Multiple Scale and Singular Perturbation Methods (Applied Mathematical Sciences, 114). Springer, New York, 1996.CrossRefGoogle Scholar
Lichtenberg, A. J. and Lieberman, M. A.. Regular and Chaotic Dynamics (Applied Mathematical Sciences, 38). Springer, New York, 1992.CrossRefGoogle Scholar
Lopes Dias, J.. A normal form theorem for Brjuno skew systems through renormalization. J. Differential Equations 230(1) (2006), 1–23.CrossRefGoogle Scholar
Meyer, K. R. and Sell, G. R.. Melnikov transforms, Bernoulli bundles, and almost periodic perturbations. Trans. Amer. Math. Soc. 314(1) (1989), 63–105.Google Scholar
Moser, J.. Combination tones for Duffing’s equation. Comm. Pure Appl. Math. 18 (1965), 167–181.CrossRefGoogle Scholar
Moser, J.. Convergent series expansions for quasi-periodic motions. Math. Ann. 169 (1967), 136–176.CrossRefGoogle Scholar
Nayfeh, A. H. and Mook, D. T.. Nonlinear Oscillations. John Wiley & Sons, New York, 1979.Google Scholar
Palmer, K. J.. Exponential dichotomies and transversal homoclinic points. J. Differential Equations 55(2) (1984), 225–256.CrossRefGoogle Scholar
Ponce, M.. On the persistence of invariant curves for fibered holomorphic transformations. Comm. Math. Phys. 289(1) (2009), 1–44.CrossRefGoogle Scholar
Procesi, M.. Exponentially small splitting and Arnold diffusion for multiple time scale systems. Rev. Math. Phys. 15(4) (2003), 339–386.CrossRefGoogle Scholar
Reitmann, V.. Dimension estimates for invariant sets of dynamical systems. Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Ed. Fiedler, B.. Springer, Berlin, 2001, pp. 585–615.CrossRefGoogle Scholar
Séré, É.. Looking for the Bernoulli shift. Ann. Inst. H. Poincaré Anal. Non Linéaire 10(5) (1993), 561–590.CrossRefGoogle Scholar
Scheurle, J.. Chaotic solutions of systems with almost periodic forcing. Z. Angew. Math. Phys. 37(1) (1986), 12–26.CrossRefGoogle Scholar
Stenlund, M.. An expansion of the homoclinic splitting matrix for the rapidly, quasiperiodically, forced pendulum. J. Math. Phys. 51(7) (2010), 072902; 40 pp.CrossRefGoogle Scholar
Treshchev, D.. Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity 17(5) (2004), 1803–1841.CrossRefGoogle Scholar
Wiggins, S.. Global Bifurcations and Chaos. Analytical Methods (Applied Mathematical Sciences, 73). Springer, New York, 1988.CrossRefGoogle Scholar
Wiggins, S.. Chaos in the quasiperiodically forced Duffing oscillator. Phys. Lett. A 124(3) (1987), 138–142.CrossRefGoogle Scholar
Xu, P. and Jing, Zh.. Quasi-periodic solutions and sub-harmonic bifurcations of Duffing’s equations with quasi-periodic perturbation. Acta Math. Appl. Sin. 15(4) (1999), 374–384.Google Scholar
Yagasaki, K.. Second-order averaging and chaos in quasiperiodically forced weakly nonlinear oscillators. Phys. D 44(3) (1990), 445–458.CrossRefGoogle Scholar