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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1007/s002200050190.  Google Scholar

[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.  doi: 10.3934/dcds.2004.11.757.  Google Scholar

[3]

A. Delshams and R. Ramírez-Ros, Poincaré-Mel'nikov-Arnol'd method for analytic planar maps,, Nonlinearity, 9 (1996), 1.  doi: 10.1088/0951-7715/9/1/001.  Google Scholar

[4]

A. Delshams and R. Ramírez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps,, J. Nonlinear Sci., 8 (1998), 317.  doi: 10.1007/s003329900054.  Google Scholar

[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).   Google Scholar

[6]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices,, Ergodic Theory Dynam. Systems, 10 (1990), 319.   Google Scholar

[7]

V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map,, Ann. Inst. Fourier (Grenoble), 51 (2001), 513.   Google Scholar

[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.   Google Scholar

[9]

V. G. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map,, Comm. Math. Phys., 201 (1999), 155.  doi: 10.1007/s002200050553.  Google Scholar

[10]

V. G. Gelfreich, V. F. Lazutkin and M. B. Tabanov, Exponentially small splittings in Hamiltonian systems,, Chaos, 1 (1991), 137.  doi: 10.1063/1.165823.  Google Scholar

[11]

V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation,, Nonlinearity, 6 (1993), 57.  doi: 10.1088/0951-7715/6/1/004.  Google Scholar

[12]

V. F. Lazutkin, Splitting of separatrices for the Chirikov standard map,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), (2003), 25.   Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[15]

E. M. McMillan, A problem in the stability of periodic systems,, In, (1971), 219.   Google Scholar

[16]

C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation,, Ann. Inst. Fourier (Grenoble), 53 (2003), 1185.   Google Scholar

[17]

F. W. J. Olver, "Asymptotics and Special Functions,", Academic Press [A subsidiary of Harcourt Brace Jovanovich, (1974).   Google Scholar

[18]

D. Sauzin, A new method for measuring the splitting of invariant manifolds,, Ann. Sci. École Norm. Sup., 34 (2001), 159.   Google Scholar

[19]

J. Scheurle, J. E. Marsden and P. Holmes, Exponentially small estimates for separatrix splittings,, In, (1991), 187.   Google Scholar

[20]

Y. B. Suris, Integrable mappings of standard type,, Funktsional. Anal. i Prilozhen., 23 (1989), 84.   Google Scholar

[21]

Y. B. Suris, On the complex separatrices of some standard-like maps,, Nonlinearity, 7 (1994), 1225.  doi: 10.1088/0951-7715/7/4/008.  Google Scholar

[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.  doi: 10.1063/1.166349.  Google Scholar

[23]

D. V. Treschev, Splitting of separatrices for a pendulum with rapidly oscillating suspension point,, Russian J. Math. Phys., 5 (1997), 63.   Google Scholar

[24]

J.-C. Yoccoz, Une erreur féconde du mathématicien Henri Poincaré,, Gaz. Math., 107 (2006), 19.   Google Scholar

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.  doi: 10.1007/s002200050190.  Google Scholar

[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.  doi: 10.3934/dcds.2004.11.757.  Google Scholar

[3]

A. Delshams and R. Ramírez-Ros, Poincaré-Mel'nikov-Arnol'd method for analytic planar maps,, Nonlinearity, 9 (1996), 1.  doi: 10.1088/0951-7715/9/1/001.  Google Scholar

[4]

A. Delshams and R. Ramírez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps,, J. Nonlinear Sci., 8 (1998), 317.  doi: 10.1007/s003329900054.  Google Scholar

[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).   Google Scholar

[6]

E. Fontich and C. Simó, Invariant manifolds for near identity differentiable maps and splitting of separatrices,, Ergodic Theory Dynam. Systems, 10 (1990), 319.   Google Scholar

[7]

V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map,, Ann. Inst. Fourier (Grenoble), 51 (2001), 513.   Google Scholar

[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.   Google Scholar

[9]

V. G. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map,, Comm. Math. Phys., 201 (1999), 155.  doi: 10.1007/s002200050553.  Google Scholar

[10]

V. G. Gelfreich, V. F. Lazutkin and M. B. Tabanov, Exponentially small splittings in Hamiltonian systems,, Chaos, 1 (1991), 137.  doi: 10.1063/1.165823.  Google Scholar

[11]

V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation,, Nonlinearity, 6 (1993), 57.  doi: 10.1088/0951-7715/6/1/004.  Google Scholar

[12]

V. F. Lazutkin, Splitting of separatrices for the Chirikov standard map,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), (2003), 25.   Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[15]

E. M. McMillan, A problem in the stability of periodic systems,, In, (1971), 219.   Google Scholar

[16]

C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation,, Ann. Inst. Fourier (Grenoble), 53 (2003), 1185.   Google Scholar

[17]

F. W. J. Olver, "Asymptotics and Special Functions,", Academic Press [A subsidiary of Harcourt Brace Jovanovich, (1974).   Google Scholar

[18]

D. Sauzin, A new method for measuring the splitting of invariant manifolds,, Ann. Sci. École Norm. Sup., 34 (2001), 159.   Google Scholar

[19]

J. Scheurle, J. E. Marsden and P. Holmes, Exponentially small estimates for separatrix splittings,, In, (1991), 187.   Google Scholar

[20]

Y. B. Suris, Integrable mappings of standard type,, Funktsional. Anal. i Prilozhen., 23 (1989), 84.   Google Scholar

[21]

Y. B. Suris, On the complex separatrices of some standard-like maps,, Nonlinearity, 7 (1994), 1225.  doi: 10.1088/0951-7715/7/4/008.  Google Scholar

[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.  doi: 10.1063/1.166349.  Google Scholar

[23]

D. V. Treschev, Splitting of separatrices for a pendulum with rapidly oscillating suspension point,, Russian J. Math. Phys., 5 (1997), 63.   Google Scholar

[24]

J.-C. Yoccoz, Une erreur féconde du mathématicien Henri Poincaré,, Gaz. Math., 107 (2006), 19.   Google Scholar

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