American Institute of Mathematical Sciences

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  2011, 18: 1-11. doi: 10.3934/era.2011.18.1

Special functions created by Borel-Laplace transform of Hénon map

Chihiro Matsuoka 1, and  Koichi Hiraide 2,

1. 

Department of Physics, Graduate school of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama 790-8577, Japan
2. 

Department of Mathematics, Graduate school of Science and Technology, Ehime University, Bunkyocho 2-5, Matsuyama 790-8577, Japan

Received  June 2010 Published  January 2011

We present a novel class of functions that can describe the stable and unstable manifolds of the Hénon map. We propose an algorithm to construct these functions by using the Borel-Laplace transform. Neither linearization nor perturbation is applied in the construction, and the obtained functions are exact solutions of the Hénon map. We also show that it is possible to depict the chaotic attractor of the map by using one of these functions without explicitly using the properties of the attractor.
Keywords: Hénon map., asymptotic expansion, Borel-Laplace transform, difference equation.
Mathematics Subject Classification: Primary: 39A45; Secondary: 44A1.
Citation: Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1
References:
[1]

M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77. doi: doi:10.1007/BF01608556.  Google Scholar

[2]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141. doi: doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[3]

V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation, Nonlinearity, 6 (1993), 57-70. doi: doi:10.1088/0951-7715/6/1/004.  Google Scholar

[4]

A. Tovbis, Asymptotics beyond all orders and analytic properties of inverse Laplace trnsforms of solutions, Commun. Math. Phys., 163 (1994), 245-255. doi: doi:10.1007/BF02102008.  Google Scholar

[5]

A. Tobvis, M. Tsuchiya and C. Jaffe, 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: doi:10.1063/1.166349.  Google Scholar

[6]

K. Nakamura and M. Hamada, Asymptotic expansion of homoclinic structures in a symplectic mapping, J. Phys. A, 29 (1996), 7315-7327. doi: 10.1088/0305-4470/29/22/025.  Google Scholar

[7]

V. F. Lazutkin, I. G. Schachmannski and M. B. Tabanov, Splitting of separatrices for standard and semistandard mappings, Physica D, 40 (1989), 235-248. doi: doi:10.1016/0167-2789(89)90065-1.  Google Scholar

[8]

M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. math., 85 (1991), 129-181.  Google Scholar

[9]

H. Segur, S. Tanveer and H. Levine (eds), "Asymptotics Beyond All Orders," (Plenum, New York), 1991. Google Scholar

[10]

A. Voros, The return of quartic oscillator: The complex WKB method, Ann. Inst. H. Poincaré 39 (1983), 211-338.  Google Scholar

[11]

J. Écalle, "Les Fonctions Résurgence vol. 1," (French) [Resurgent functions. Vol. I] Les algèbres de fonctions résurgentes. [The algebras of resurgent functions] With an English foreword. Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 5. Université de Paris-Sud, Département de Mathématique, Orsay, 1981.  Google Scholar

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J. Écalle, "Les Fonctions Résurgence vol. 2," (French) [Resurgent functions. Vol. II] Les fonctions résurgentes appliquées à l'itération. [Resurgent functions applied to iteration] Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 6. Université de Paris-Sud, Département de Mathématique, Orsay, 1981.  Google Scholar

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J. Écalle, "Les Fonctions Résurgence vol. 3," (French) [Resurgent functions. Vol. III] L' équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects] Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 85-5. Université de Paris-Sud, Département de Math¨¦matiques, Orsay, 1985.  Google Scholar

[14]

B. Y. Sternin and V. E. Shatalov, "Borel-Laplace Transform and Asymptotic Theory," Introduction to Resurgent Analysis, CRC Press, Boca Raton, FL, 1996. Google Scholar

[15]

V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier (Grenoble), 51 (2001), 513-67.  Google Scholar

[16]

S. Newhouse and T. Pignataro, On the estimation of topological entropy, J. Stat. Phys., 72 (1993), 1331-1351.  Google Scholar

show all references

References:
[1]

M. Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77. doi: doi:10.1007/BF01608556.  Google Scholar

[2]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci., 20 (1963), 130-141. doi: doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[3]

V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation, Nonlinearity, 6 (1993), 57-70. doi: doi:10.1088/0951-7715/6/1/004.  Google Scholar

[4]

A. Tovbis, Asymptotics beyond all orders and analytic properties of inverse Laplace trnsforms of solutions, Commun. Math. Phys., 163 (1994), 245-255. doi: doi:10.1007/BF02102008.  Google Scholar

[5]

A. Tobvis, M. Tsuchiya and C. Jaffe, 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: doi:10.1063/1.166349.  Google Scholar

[6]

K. Nakamura and M. Hamada, Asymptotic expansion of homoclinic structures in a symplectic mapping, J. Phys. A, 29 (1996), 7315-7327. doi: 10.1088/0305-4470/29/22/025.  Google Scholar

[7]

V. F. Lazutkin, I. G. Schachmannski and M. B. Tabanov, Splitting of separatrices for standard and semistandard mappings, Physica D, 40 (1989), 235-248. doi: doi:10.1016/0167-2789(89)90065-1.  Google Scholar

[8]

M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. math., 85 (1991), 129-181.  Google Scholar

[9]

H. Segur, S. Tanveer and H. Levine (eds), "Asymptotics Beyond All Orders," (Plenum, New York), 1991. Google Scholar

[10]

A. Voros, The return of quartic oscillator: The complex WKB method, Ann. Inst. H. Poincaré 39 (1983), 211-338.  Google Scholar

[11]

J. Écalle, "Les Fonctions Résurgence vol. 1," (French) [Resurgent functions. Vol. I] Les algèbres de fonctions résurgentes. [The algebras of resurgent functions] With an English foreword. Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 5. Université de Paris-Sud, Département de Mathématique, Orsay, 1981.  Google Scholar

[12]

J. Écalle, "Les Fonctions Résurgence vol. 2," (French) [Resurgent functions. Vol. II] Les fonctions résurgentes appliquées à l'itération. [Resurgent functions applied to iteration] Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 6. Université de Paris-Sud, Département de Mathématique, Orsay, 1981.  Google Scholar

[13]

J. Écalle, "Les Fonctions Résurgence vol. 3," (French) [Resurgent functions. Vol. III] L' équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects] Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 85-5. Université de Paris-Sud, Département de Math¨¦matiques, Orsay, 1985.  Google Scholar

[14]

B. Y. Sternin and V. E. Shatalov, "Borel-Laplace Transform and Asymptotic Theory," Introduction to Resurgent Analysis, CRC Press, Boca Raton, FL, 1996. Google Scholar

[15]

V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier (Grenoble), 51 (2001), 513-67.  Google Scholar

[16]

S. Newhouse and T. Pignataro, On the estimation of topological entropy, J. Stat. Phys., 72 (1993), 1331-1351.  Google Scholar

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