July  2013, 33(7): 2829-2859. doi: 10.3934/dcds.2013.33.2829

Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom

1. 

School of Mathematics, Institute for Advanced Study, Einstein Drive, Simonyi Hall, Princeton, New Jersey, 08540, United States

Received  March 2012 Revised  April 2012 Published  January 2013

In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems generically appear hyperbolic periodic orbits. We study the possible transversal intersections of their invariant manifolds, which is exponentially small, and we give an asymptotic formula for the measure of the splitting. We see that its asymptotic first order is of the form $K \varepsilon^{\beta} \text{e}^{-a/\varepsilon}$ and we identify the constants $K,\beta,a$ in terms of the system features. We compare our results with the classical Melnikov Theory and we show that, typically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants $K$ and $\beta$ involved in the formula.
Citation: Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829
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show all references

References:
[1]

3 of Encyclopaedia Math. Sci., Springer, Berlin, 1988.  Google Scholar

[2]

Russian Math. Surveys, 18 (1963), 9-36.  Google Scholar

[3]

Nonlinearity, 19 (2006), 1415-1445. doi: 10.1088/0951-7715/19/6/011.  Google Scholar

[4]

Mem. Amer. Math. Soc., 167 (2004), x-83.  Google Scholar

[5]

J. Differential Equations, 210 (2005), 106-134. doi: 10.1016/j.jde.2004.10.017.  Google Scholar

[6]

preprint, arXiv:1201.5152, 2011. doi: 10.1016/j.jde.2012.09.003.  Google Scholar

[7]

Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144 pp.  Google Scholar

[8]

Comm. Math. Phys., 189 (1997), 35-71. doi: 10.1007/s002200050190.  Google Scholar

[9]

Discrete Contin. Dyn. Syst., 11 (2004), 785-826. doi: 10.3934/dcds.2004.11.785.  Google Scholar

[10]

Comm. Math. Phys., 150 (1992), 433-463.  Google Scholar

[11]

Math. Phys. Electron. J., 3 (1997), 40 pp. (electronic).  Google Scholar

[12]

Nonlinear Anal., 20 (1993), 733-744. doi: 10.1016/0362-546X(93)90031-M.  Google Scholar

[13]

J. Differential Equations, 119 (1995), 310-335. doi: 10.1006/jdeq.1995.1093.  Google Scholar

[14]

in "Seminar on Dynamical Systems (St. Petersburg, 1991)" 12 of Progr. Nonlinear Differential Equations Appl., 47-67. Birkhäuser, Basel, (1994).  Google Scholar

[15]

Phys. D, 101 (1997), 227-248. doi: 10.1016/S0167-2789(96)00133-9.  Google Scholar

[16]

Nonlinearity, 10 (1997), 175-193. doi: 10.1088/0951-7715/10/1/012.  Google Scholar

[17]

Russ. J. Math. Phys., 7 (2000), 48-71.  Google Scholar

[18]

Comm. Math. Phys., 202 (1999), 197-236. doi: 10.1007/s002200050579.  Google Scholar

[19]

Springer-Verlag, 1983.  Google Scholar

[20]

J. Nonlinear Sci., 20 (2010), 595-685. doi: 10.1007/s00332-010-9068-8.  Google Scholar

[21]

Nonlinearity, 25 (2012), 1367-1412. doi: 10.1088/0951-7715/25/5/1367.  Google Scholar

[22]

in "Hamiltonian Dynamical Systems" 81 of Contemp. Math.. (1988). doi: 10.1090/conm/081/986267.  Google Scholar

[23]

Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.  Google Scholar

[24]

Mem. Amer. Math. Soc., 163 (2003), viii+145.  Google Scholar

[25]

Trans. Moscow Math. Soc., 12 (1963), 1-57.  Google Scholar

[26]

Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, (1962), 1-20.  Google Scholar

[27]

Prikl. Mat. Mekh., 48 (1984), 197-204. doi: 10.1016/0021-8928(84)90078-9.  Google Scholar

[28]

Ph.D thesis, Universitat Politècnica de Catalunya, 2006. Google Scholar

[29]

in "Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002)" 53 (2003), 1185-1235.  Google Scholar

[30]

Acta Mathematica, 13 (1890), 1-270. Google Scholar

[31]

Ann. Ins. Fourier, 45 (1995), 453-511.  Google Scholar

[32]

Ann. Sci. École Norm. Sup., 34 (2001), 159-221. doi: 10.1016/S0012-9593(00)01063-6.  Google Scholar

[33]

in "Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse)" 63-80. Princeton Univ. Press, Princeton, N.J., (1965).  Google Scholar

[34]

in "Asymptotics Beyond All Orders (La Jolla, CA, 1991)" 284 of NATO Adv. Sci. Inst. Ser. B Phys., 187-195. Plenum, New York, (1991).  Google Scholar

[35]

Nonlinearity, 22 (2009), 1191-1245. doi: 10.1088/0951-7715/22/5/012.  Google Scholar

[36]

Russ. J. Math. Phys., 5 (1997), 63-98.  Google Scholar

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