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Exponentially small splitting of separatrices in the perturbed McMillan map
1. | Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Ed-C3, Jordi Girona, 1-3, 08034 Barcelona |
2. | IMCCE, Observatoire de Paris, 77 Avenue Denfert-Rochereau, 75014 Paris |
3. | Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona |
References:
[1] |
A. Delshams, V. Gelfreich, À. Jorba and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comm. Math. Phys., 189 (1987), 35-71.
doi: 10.1007/s002200050190. |
[2] |
A. Delshams and P. Gutiérrez, Exponentially small spliting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits, Discrete Contin. Dyn. Syst., 11 (2004), 757-783.
doi: 10.3934/dcds.2004.11.757. |
[3] |
A. Delshams and R. Ramírez-Ros, Poincaré-Mel'nikov-Arnol'd method for analytic planar maps, Nonlinearity, 9 (1996), 1-26.
doi: 10.1088/0951-7715/9/1/001. |
[4] |
A. Delshams and R. Ramírez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps, J. Nonlinear Sci., 8 (1998), 317-352.
doi: 10.1007/s003329900054. |
[5] |
A. Delshams and T. M. Seara, Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom, Math. Phys. Electron. J., 3 (1997), 40 pp. |
[6] |
E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergodic Theory Dynam. Systems, 10 (1990), 319-346. |
[7] |
V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier (Grenoble), 51 (2001), 513-567. |
[8] |
V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 511-536. |
[9] |
V. G. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys., 201 (1999), 155-216.
doi: 10.1007/s002200050553. |
[10] |
V. G. Gelfreich, V. F. Lazutkin and M. B. Tabanov, Exponentially small splittings in Hamiltonian systems, Chaos, 1 (1991), 137-142.
doi: 10.1063/1.165823. |
[11] |
V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation, Nonlinearity, 6 (1993), 57-70.
doi: 10.1088/0951-7715/6/1/004. |
[12] |
V. F. Lazutkin, Splitting of separatrices for the Chirikov standard map, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (Teor. Predst. Din. Sist. Spets. Vyp. 8), 25-55, 285, 2003. Translated from the Russian and with a preface by V. Gelfreich, Workshop on Differential Equations (Saint-Petersburg, 2002). |
[13] |
P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems, Mem. Amer. Math. Soc., 163 (2003), viii+145. |
[14] |
P. Martín, D. Sauzin and T. M. Seara, Resurgence of inner solutions for perturbations of the McMillan map, Discrete Contin. Dyn. Syst., 31 (2011), 165-207. |
[15] |
E. M. McMillan, A problem in the stability of periodic systems, In "Topics in Modern Physics: A Tribute to EU Condon" (W.E. Brittin and Odeabasih eds.), pages 219-244. Colorado Associated University Press, Boulder, CO, 1971. |
[16] |
C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation, Ann. Inst. Fourier (Grenoble), 53 (2003), 1185-1235. |
[17] |
F. W. J. Olver, "Asymptotics and Special Functions," Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. |
[18] |
D. Sauzin, A new method for measuring the splitting of invariant manifolds, Ann. Sci. École Norm. Sup., 34 (2001), 159-221. |
[19] |
J. Scheurle, J. E. Marsden and P. Holmes, Exponentially small estimates for separatrix splittings, In "Asymptotics Beyond All Orders" (La Jolla, CA, 1991), volume 284 of NATO Adv. Sci. Inst. Ser. B Phys., pages 187-195. Plenum, New York, 1991. |
[20] |
Y. B. Suris, Integrable mappings of standard type, Funktsional. Anal. i Prilozhen., 23 (1989), 84-85. |
[21] |
Y. B. Suris, On the complex separatrices of some standard-like maps, Nonlinearity, 7 (1994), 1225-1236.
doi: 10.1088/0951-7715/7/4/008. |
[22] |
A. Tovbis, M. Tsuchiya and C. Jaffé, Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Hénon map as an example, Chaos, 8 (1998), 665-681.
doi: 10.1063/1.166349. |
[23] |
D. V. Treschev, Splitting of separatrices for a pendulum with rapidly oscillating suspension point, Russian J. Math. Phys., 5 (1997), 63-98. |
[24] |
J.-C. Yoccoz, Une erreur féconde du mathématicien Henri Poincaré, Gaz. Math., 107 (2006), 19-26. |
show all references
References:
[1] |
A. Delshams, V. Gelfreich, À. Jorba and T. M. Seara, Exponentially small splitting of separatrices under fast quasiperiodic forcing, Comm. Math. Phys., 189 (1987), 35-71.
doi: 10.1007/s002200050190. |
[2] |
A. Delshams and P. Gutiérrez, Exponentially small spliting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits, Discrete Contin. Dyn. Syst., 11 (2004), 757-783.
doi: 10.3934/dcds.2004.11.757. |
[3] |
A. Delshams and R. Ramírez-Ros, Poincaré-Mel'nikov-Arnol'd method for analytic planar maps, Nonlinearity, 9 (1996), 1-26.
doi: 10.1088/0951-7715/9/1/001. |
[4] |
A. Delshams and R. Ramírez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps, J. Nonlinear Sci., 8 (1998), 317-352.
doi: 10.1007/s003329900054. |
[5] |
A. Delshams and T. M. Seara, Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom, Math. Phys. Electron. J., 3 (1997), 40 pp. |
[6] |
E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices, Ergodic Theory Dynam. Systems, 10 (1990), 319-346. |
[7] |
V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier (Grenoble), 51 (2001), 513-567. |
[8] |
V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 511-536. |
[9] |
V. G. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys., 201 (1999), 155-216.
doi: 10.1007/s002200050553. |
[10] |
V. G. Gelfreich, V. F. Lazutkin and M. B. Tabanov, Exponentially small splittings in Hamiltonian systems, Chaos, 1 (1991), 137-142.
doi: 10.1063/1.165823. |
[11] |
V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation, Nonlinearity, 6 (1993), 57-70.
doi: 10.1088/0951-7715/6/1/004. |
[12] |
V. F. Lazutkin, Splitting of separatrices for the Chirikov standard map, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (Teor. Predst. Din. Sist. Spets. Vyp. 8), 25-55, 285, 2003. Translated from the Russian and with a preface by V. Gelfreich, Workshop on Differential Equations (Saint-Petersburg, 2002). |
[13] |
P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems, Mem. Amer. Math. Soc., 163 (2003), viii+145. |
[14] |
P. Martín, D. Sauzin and T. M. Seara, Resurgence of inner solutions for perturbations of the McMillan map, Discrete Contin. Dyn. Syst., 31 (2011), 165-207. |
[15] |
E. M. McMillan, A problem in the stability of periodic systems, In "Topics in Modern Physics: A Tribute to EU Condon" (W.E. Brittin and Odeabasih eds.), pages 219-244. Colorado Associated University Press, Boulder, CO, 1971. |
[16] |
C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation, Ann. Inst. Fourier (Grenoble), 53 (2003), 1185-1235. |
[17] |
F. W. J. Olver, "Asymptotics and Special Functions," Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. |
[18] |
D. Sauzin, A new method for measuring the splitting of invariant manifolds, Ann. Sci. École Norm. Sup., 34 (2001), 159-221. |
[19] |
J. Scheurle, J. E. Marsden and P. Holmes, Exponentially small estimates for separatrix splittings, In "Asymptotics Beyond All Orders" (La Jolla, CA, 1991), volume 284 of NATO Adv. Sci. Inst. Ser. B Phys., pages 187-195. Plenum, New York, 1991. |
[20] |
Y. B. Suris, Integrable mappings of standard type, Funktsional. Anal. i Prilozhen., 23 (1989), 84-85. |
[21] |
Y. B. Suris, On the complex separatrices of some standard-like maps, Nonlinearity, 7 (1994), 1225-1236.
doi: 10.1088/0951-7715/7/4/008. |
[22] |
A. Tovbis, M. Tsuchiya and C. Jaffé, Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Hénon map as an example, Chaos, 8 (1998), 665-681.
doi: 10.1063/1.166349. |
[23] |
D. V. Treschev, Splitting of separatrices for a pendulum with rapidly oscillating suspension point, Russian J. Math. Phys., 5 (1997), 63-98. |
[24] |
J.-C. Yoccoz, Une erreur féconde du mathématicien Henri Poincaré, Gaz. Math., 107 (2006), 19-26. |
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