Exponentially small splitting of separatrices in a weakly hyperbolic case. (English) Zbl 1122.37046
Summary: We validate the Poincaré-Melnikov method in the singular case of high-frequency periodic perturbations of the Hamiltonian \(h_0(x,y)=(1/2)y^{2}-x^{3}+x^{4}\) under appropriate conditions, which among other things, imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one. The splitting is exponentially small.
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 34D35 | Stability of manifolds of solutions to ordinary differential equations |
| 37J30 | Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria) |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 70H07 | Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics |
References:
| [1] | S. Angenent, A variational interpretation of Melnikov’s function and exponentially small separatrix splitting, Symplectic Geometry, London Mathematical Society, Lecture Note Series, vol. 192, Cambridge University Press, Cambridge, 1993, pp. 5-35.; S. Angenent, A variational interpretation of Melnikov’s function and exponentially small separatrix splitting, Symplectic Geometry, London Mathematical Society, Lecture Note Series, vol. 192, Cambridge University Press, Cambridge, 1993, pp. 5-35. · Zbl 0810.34037 |
| [2] | Arnold, V. I., Instability of dynamical systems with several degrees of freedom, Soviet Math. Dokl., 5, 581-585 (1964) · Zbl 0135.42602 |
| [3] | Baldomá, I.; Fontich, E., Exponentially small splitting of invariant manifolds of parabolic points, Mem. Amer. Math. Soc., 167, 792, x+83 (2004) · Zbl 1040.37050 |
| [4] | Casasayas, J.; Fontich, E.; Nunes, A., Homoclinic orbits to parabolic points, NoDEA Nonlinear Differential Equations Appl., 4, 201-216 (1997) · Zbl 0920.70017 |
| [5] | Delshams, A.; Ramírez-Ros, R., Poincaré-Melnikov-Arnold method for analytic planar maps, Nonlinearity, 9, 1-26 (1996) · Zbl 0887.58029 |
| [6] | Delshams, A.; Ramírez-Ros, R., Exponentially small splitting of separatrices for perturbed integrable standard-like maps, J. Nonlinear Sci., 8, 317-352 (1998) · Zbl 0904.58050 |
| [7] | Delshams, A.; Seara, T. M., An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum, Comm. Math. Phys., 150, 433-463 (1992) · Zbl 0765.70016 |
| [8] | Delshams, A.; Seara, T. M., Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom, Math. Phys. Electron. J., 3, 4-40 (1997) · Zbl 0891.58035 |
| [9] | Easton, R. W., Computing the dependence on a parameter of a family of unstable manifoldsgeneralized Melnikov formulas, Nonlinear Anal., 8, 1-4 (1984) · Zbl 0536.58027 |
| [10] | Fiedler, B.; Scheurle, J., Discretization of homoclinic orbits, rapid forcing and “homoclinic” chaos, Mem. Amer. Math. Soc., 119, 570, viii+79 (1996) · Zbl 0923.34049 |
| [11] | Fontich, E., Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations, Nonlinear Anal., 20, 733-744 (1993) · Zbl 0773.34042 |
| [12] | Fontich, E., Rapidly forced planar vector fields and splitting of separatrices, J. Differential Equations, 119, 310-335 (1995) · Zbl 0827.34041 |
| [13] | Fontich, E.; Simó, C., The splitting of separatrices for analytic diffeomorphisms, Ergodic Theory Dynamical Systems, 10, 295-318 (1990) · Zbl 0706.58061 |
| [14] | V.G. Gelfreich, Separatrices splitting for the rapidly forced pendulum, Seminar on Dynamical Systems (St. Petersburg), Progress in Nonlinear Differential Equations and their Applications, vol. 12, Birkhäuser, Basel, 1991, pp. 47-67.; V.G. Gelfreich, Separatrices splitting for the rapidly forced pendulum, Seminar on Dynamical Systems (St. Petersburg), Progress in Nonlinear Differential Equations and their Applications, vol. 12, Birkhäuser, Basel, 1991, pp. 47-67. · Zbl 0792.34053 |
| [15] | Gelfreich, V. G., Melnikov method and exponentially small splitting of separatrices, Physica D, 101, 227-248 (1997) · Zbl 0896.70011 |
| [16] | Gelfreich, V. G., A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys., 201, 155-216 (1999) · Zbl 1042.37044 |
| [17] | Gelfreich, V. G.; Lazutkin, V. F., Splitting of separatricesperturbation theory and exponential smallness, Russian Math. Surveys, 56, 499-558 (2001) · Zbl 1071.37039 |
| [18] | Holmes, P.; Marsden, J.; Scheurle, J., Exponentially small splitting of separatrices with applications to KAM theory and degenerate bifurcations, Contemp. Math., 81, 213-244 (1988) · Zbl 0685.70017 |
| [19] | V.F. Lazutkin, Splitting of separatrices for the Chirikov’s standard map, VINITI, vol. 6372(84) 1984 (english version in Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 300 (2003), Teor. Predst. Din. Sist. Spets. Vyp. 8, 25-55, 285).; V.F. Lazutkin, Splitting of separatrices for the Chirikov’s standard map, VINITI, vol. 6372(84) 1984 (english version in Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 300 (2003), Teor. Predst. Din. Sist. Spets. Vyp. 8, 25-55, 285). |
| [20] | Melnikov, V. F., On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12, 3-56 (1963) · Zbl 0135.31001 |
| [21] | Neishtadt, A. I., The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48, 133-139 (1984) · Zbl 0571.70022 |
| [22] | Poincaré, H., Sur le problème des trois corps et les équations de la dynamique, Acta Math., 13, 1-271 (1890) · JFM 22.0907.01 |
| [23] | J. Scheurle, J. Marsden, P. Holmes, Exponentially small estimates for separatrix splitting, Asymptotics Beyond all Orders (La Jolla, CA, 1991), NATO Advanced Science Institutes Series B: Physics, vol. 284, Plenum, New York, 1991, pp. 187-195.; J. Scheurle, J. Marsden, P. Holmes, Exponentially small estimates for separatrix splitting, Asymptotics Beyond all Orders (La Jolla, CA, 1991), NATO Advanced Science Institutes Series B: Physics, vol. 284, Plenum, New York, 1991, pp. 187-195. |
| [24] | Treschev, D. V., Separatrix splitting for a pendulum with rapidly oscillating suspension point, Russian J. Math. Phys., 5, 63-98 (1997) · Zbl 0947.34034 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
