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
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

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

[1]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406

[2]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002

[3]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019

[4]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001

[5]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029

[6]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[7]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407

[8]

G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010

[9]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040

[10]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

[11]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[12]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[13]

V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511

[14]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[15]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[16]

Wei Wang, Degen Huang, Haitao Yu. Word sense disambiguation based on stretchable matching of the semantic template. Mathematical Foundations of Computing, 2021, 4 (1) : 1-13. doi: 10.3934/mfc.2020022

[17]

Alexander Dabrowski, Ahcene Ghandriche, Mourad Sini. Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies. Inverse Problems & Imaging, 2021, 15 (3) : 555-597. doi: 10.3934/ipi.2021005

[18]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[19]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377

[20]

Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021047

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]