Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps. (English) Zbl 0942.37016

The author deals with the splitting of a small separatrix loop that appears after the Bogdanov-Takens (saddle-center) bifurcation in an analytic family of area-preserving maps and proves that in general the splitting is described by almost the “same formula” as in the case of the standard map. More precisely, the author derives an asymptotic formula that describes the splitting, and studies the properties of the pre-exponential factor.


37D30 Partially hyperbolic systems and dominated splittings
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI


[1] Broer, H.; Roussarie, R.; Simó, C., Invariant circles in the bogdanov – takens bifurcation for diffeomorphisms, Ergodic theory dynam. systems, 16, 6, 1147-1172, (1996) · Zbl 0876.58032
[2] Chernov, V., On separatrix splitting of some quadratic area-preserving maps of the plane, Regular & chaotic dynam., 3, 1, 49-65, (1998) · Zbl 0924.58065
[3] Chirikov, B.V., A universal instability of many-dimensional oscillator systems, Phys. rep., 52, 263-379, (1979)
[4] Delshams, A.; Ramírez-Ros, R., Poincaré-melnikov – arnold method for analytic planar maps, Nonlinearity, 9, 1-26, (1996) · Zbl 0887.58029
[5] Delshams, A.; Ramírez-Ros, R., Exponentially small splitting of separatrices for perturbed integrable standard-like maps, J. nonlinear sci., 8, 3, 317-352, (1998) · Zbl 0904.58050
[6] A. Delshams, R. Ramírez-Ros, Singular separatrix splitting and Melnikov method: an experimental study, mp_arc@math.utexas.edu, preprint, Experiment. Math. (1999) 98-323.
[7] J. Ecalle, Les fonctions résurgentes, vol. 2, Publ. Math. d’Orsay, Paris, 1981. · Zbl 0499.30034
[8] V.G. Gelfreich, Separatrices splitting for polymomial area-preserving maps, in: M. Sh. Birman (Ed.), Topics in Math. Phys. (Russian), vol. 13, Leningrad State University, 1991, pp. 108-116.
[9] Gelfreich, V.G., A proof of the exponentially small transversality of the separatrices for the standard map, Commun. math. phys., 201, 1, 155-216, (1999) · Zbl 1042.37044
[10] Gelfreich, V.G.; Lazutkin, V.F.; Svanidze, N.V., A refined formula for the separatrix splitting for the standard map, Physica D, 71, 2, 82-101, (1994) · Zbl 0812.70017
[11] Gelfreich, V.G.; Lazutkin, V.F.; Tabanov, M.B., Exponentially small splitting in Hamiltonian systems, Chaos, 1, 2, 137-142, (1991) · Zbl 0899.58016
[12] V. Gelfreich, D. Sauzin, Borel summation and the splitting of separatrices for the Henon map, Notes Scientifuqes et techniques du Bureau des Longitudes, S067, Mai 1999, p. 48. · Zbl 0988.37031
[13] Glasser, M.L.; Papageorgiou, V.G.; Bountis, T.C., Melnikov’s function for two-dimensional mappings, SIAM J. appl. math., 49, 3, 692-703, (1989) · Zbl 0687.58023
[14] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983. · Zbl 0515.34001
[15] V.F. Lazutkin, Splitting of separatrices for the Chirikov’s standard map (Russian), VINITI no. 6372/84 (1984).
[16] Lazutkin, V.F., An analytic integral along the separatrix of the semistandard map: existence and an exponential estimate for the distance between the stable and unstable separatrices, St.-Petersburg math. J., 4, 4, 721-748, (1993) · Zbl 0791.58036
[17] Lazutkin, V.F.; Schachmanski, I.G.; Tabanov, M.B., Splitting of separatrices for standard and semistandard mappings, Physica D, 40, 235-348, (1989) · Zbl 0825.58033
[18] Melnikov, V.K., On the stability of the center for time periodic perturbations, Trans. Moscow math. soc., 12, 3-56, (1963)
[19] Neishtadt, A.I., The separation of motion in systems with rapidly rotating phase, J. appl. math. mech., 48, 2, 133-139, (1984)
[20] Suris, Yu.B., On the complex separatrices of some standard-like maps, Nonlinearity, 7, 4, 1225-1236, (1994) · Zbl 0813.58024
[21] Treshchev, D.V., An averaging method for Hamiltonian systems, exponentially close to integrable ones, Chaos, 6, 1, 6-14, (1996) · Zbl 1055.70509
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.