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Separatrix splitting at a Hamiltonian \(0^2i\omega\) bifurcation. (English) Zbl 1347.37102
The authors consider in a neighborhood of the origin in \(\mathbb{R}^4\) the generic bifurcation of the equilibrium in one-parameter Hamiltonian systems with two degrees of freedom and relevant real-analytic families of Hamiltonians \(F_{\mu}=H_{\mu}(x_1,x_2,y_1,y_2)\) under the assumption that the unperturbed fixed point zero has two pure imaginary eigenvalues \(\pm i \omega_0\) and the non-semisimple double zero eigenvalue. For this example it is studied the splitting of a separatrix in a generic unfolding of a degenerate equilibrium. The corresponding integrable normal forms has a normally elliptic invariant manifold of dimension two on which the truncated normal form has a separatrix loop. The splitting of this loop is exponentially small compared to the small parameter.
The basic result of this article consists in the derivation of the asymptotic expression for the separatrix splitting.

MSC:
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
70K50 Bifurcations and instability for nonlinear problems in mechanics
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
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References:
[1] Alfimov, G L; Eleonsky, VM; Kulagin, NE, Dynamical systems in the theory of solitons in the presence of nonlocal interactions, Chaos, 2, 565-570, (1992) · Zbl 1055.37580
[2] Baldomá, I; Seara, TM, Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity, J. Nonlinear Sci., 16, 543-582, (2006) · Zbl 1130.37380
[3] Baldomá, I; Fontich, E; Guàrdia, M; Seara, T M, Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results, J. Differential Equations, 253, 3304-3439, (2012) · Zbl 1271.34050
[4] Baldomá, I; Castejón, O; Seara, TM, Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity, J. Dynam. Differential Equations, 25, 335-392, (2013) · Zbl 1269.37025
[5] Broer, H W; Chow, S-N; Kim, Y; Vegter, G, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys., 44, 389-432, (1993) · Zbl 0805.58047
[6] Champneys, AR, Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Phys. D, 112, 158-86, (1998) · Zbl 1194.37154
[7] Champneys, AR, Codimension-one persistence beyond all orders of homoclinic orbits to singular saddle centres in reversible systems, Nonlinearity, 14, 87-112, (2001) · Zbl 0985.37045
[8] Gelfreich, V G, Separatrix splitting for a high-frequency perturbation of the pendulum, Russ. J. Math. Phys., 7, 48-71, (2000) · Zbl 1066.34059
[9] Gelfreich, V G, Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps, Phys. D, 136, 266-279, (2000) · Zbl 0942.37016
[10] Gelfreich, V; Lazutkin, V, Splitting of separatrices: perturbation theory and exponential smallness, Russian Math. Surveys, 56, 499-558, (2001) · Zbl 1071.37039
[11] Gelfreich, V, Near strongly resonant periodic orbits in a Hamiltonian system, Proc. Natl. Acad. Sci. USA, 99, 13975-13979, (2002) · Zbl 1072.37044
[12] Gelfreich, V G; Lerman, L M, Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system, Nonlinearity, 15, 447-457, (2002) · Zbl 1001.37050
[13] Gelfreich, VG; Lerman, LM, Long-periodic orbits and invariant tori in a singularly perturbed Hamiltonian system, Phys. D, 176, 125-146, (2003) · Zbl 1008.37036
[14] Gelfreich, V; Simó, C, High-precision computations of divergent asymptotic series and homoclinic phenomena, Discrete Contin. Dyn. Syst. Ser. B, 10, 511-536, (2008) · Zbl 1169.37013
[15] Giorgilli, A, Unstable equilibria of Hamiltonian systems, Discrete Contin. Dynam. Systems, 7, 855-871, (2001) · Zbl 1016.37031
[16] Grotta Ragazzo, C, Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers, Comm. Pure Appl. Math., 50, 105-147, (1997) · Zbl 0884.58043
[17] Haragus, M. and Iooss, G., Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, London: Springer, 2011. · Zbl 1230.34002
[18] Iooss, G. and Adelmeyer, M., Topics in Bifurcation Theory and Applications, Adv. Ser. Nonlinear Dynam., vol. 3, River Edge, N.J.: World Sci., 1992. · Zbl 0833.34001
[19] Iooss, G; Lombardi, E, Polynomial normal forms with exponentially small remainder for analytic vector fields, J. Differential Equations, 212, 1-61, (2005) · Zbl 1072.34039
[20] Iooss, G; Lombardi, E, Normal forms with exponentially small remainder: application to homoclinic connections for the reversible 0\^{2}+ resonance, C. R. Math. Acad. Sci. Paris, Ser. 1, 339, 831-838, (2004) · Zbl 1066.34041
[21] Jézéquel, T., Bernard, P., and Lombardi, E., Homoclinic Connections with Many Loops near a 0\^{2} Resonant Fixed Point for Hamiltonian Systems, arXiv:1401.1509 (2014), 79 pp.
[22] Koltsova, O, Families ofmulti-round homoclinic and periodic orbits near a saddle-center equilibrium, Regul. Chaotic Dyn., 8, 191-200, (2003) · Zbl 1112.37319
[23] Koltsova, O; Lerman, LM, Periodic and homoclinic orbits in a two-parameter unfolding of a Hamiltonian system with a homoclinic orbit to a saddle-center, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5, 397-408, (1995) · Zbl 0885.58025
[24] Koltsova, OYu; Lerman, L M, Families of transverse Poincaré homoclinic orbits in 2\(N\)dimensional Hamiltonian systems close to the system with a loop to a saddle-center, Int. J. Bifurcation & Chaos, 6, 991-1006, (1996) · Zbl 0874.58018
[25] Lerman, LM; Gelfreich, VG, Fast-slow Hamiltonian dynamics near a ghost separatrix loop, J. Math. Sci. (N. Y.), 126, 1445-1466, (2005) · Zbl 1093.37025
[26] Lerman, LM, Hamiltonian systems with a separatrix loop of a saddle-center, Selecta Math. (N. S.), 10, 297-306, (1991)
[27] Llibre, J; Martínez, R; Simó, C, Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near \(L\)2 in the restricted three-body problem, J. Differential Equations, 58, 104-156, (1985) · Zbl 0594.70013
[28] Lombardi, E., Oscillatory Integrals and Phenomena beyond All Algebraic Orders: With Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Math., vol. 1741, Berlin: Springer, 2000. · Zbl 0959.34002
[29] Mielke, A; Holmes, P; O’Reiley, O, Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center, J. Dynam. Differential Equations, 4, 95-126, (1992) · Zbl 0749.58022
[30] Moser, J, On the generalization of a theorem of A. liapounoff, Comm. Pure Appl. Math., 11, 257-271, (1958) · Zbl 0082.08003
[31] Treschev, D, Splitting of separatrices for a pendulum with rapidly oscillating suspension point, Russian J. Math. Phys., 5, 63-98, (1997) · Zbl 0947.34034
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