Exponentially small splitting of separatrices in the perturbed McMillan map

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June 2011, 31(2): 301-372. doi: 10.3934/dcds.2011.31.301

Exponentially small splitting of separatrices in the perturbed McMillan map

Pau Martín ^1, , David Sauzin ^2, and Tere M. Seara ^3,

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

Received January 2010 Revised February 2011 Published June 2011

Abstract
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The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coefficients involved in the expansion have a resurgent origin, as shown in [14].

Keywords: exponentially small phenomena, splitting of separatrices, asymptotic formula., McMillan map.

Mathematics Subject Classification: 37J10, 34C37, 37C29, 34E10, 34M3.

Citation: Pau Martín, David Sauzin, Tere M. Seara. Exponentially small splitting of separatrices in the perturbed McMillan map. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 301-372. doi: 10.3934/dcds.2011.31.301

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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