Abstract
We study the existence of transverse homoclinic orbits in a singular or weakly hyperbolic Hamiltonian with three degrees of freedom as a model for the behavior of a nearly integrable Hamiltonian near a simple resonance. The considered example consists of an integrable Hamiltonian having a two-dimensional hyperbolic invariant torus with fast frequencies \(w/\sqrt \in\) and coincident whiskers or separatrices, plus a perturbation of order μ = εp giving rise to an exponentially small splitting of separatrices. We show that asymptotic estimates for the transversality of the intersections can be obtained if ω satisfies certain arithmetic properties. More precisely, we assume that ω is a quadratic vector (i.e., the frequency ratio is a quadratic irrational number) and generalizes good arithmetic properties of the golden vector. We provide a sufficient condition on the quadratic vector ω ensuring that the Poincare-Melnikov method (used for the golden vector in a previous work) can be applied to establish the existence of transverse homoclinic orbits and, in a more restricted case, their continuation for all values of ε → 0. Bibliography: 22 titles.
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REFERENCES
G. Benettin, A. Carati, and G. Gallavotti, “A rigorous implementation of the Jeans-Landau-Teller approximation for adiabatic invariants,” Nonlinearity, 10, 479–505 (1997).
B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep., 52, No.5, 263–379 (1979).
A. Delshams and P. Gutierrez, “Splitting potential and the Poincare-Melnikov method for whiskered tori in Hamiltonian systems,” J. Nonlinear Sci., 10, No.4, 433–476 (2000).
A. Delshams and P. Gutierrez, “Splitting and Melnikov potentials in Hamiltonian systems,” in: Hamiltonian Systems and Celestial Mechanics (HAMSYS-98). Proceedings of the III International Symposium (Mexico, 1998), J. Delgado, E. A. Lacomba, E. Perez-Chavela, and J. Llibre (eds.), World Sci. Publishing, Singapore (2000), pp. 111–137.
A. Delshams and P. Gutierrez, “Homoclinic orbits to invariant tori in Hamiltonian systems,” in: Multiple-Time-Scale Dynamical Systems, C. K. R. T. Jones and A. I. Khibnik (eds.), Springer-Verlag, New York (2001), pp. 1–27.
A. Delshams and P. Gutierrez, “Exponentially small splitting for whiskered tori in Hamiltonian systems: Continuation of transverse homoclinic orbits,” Mathematical Physics Preprint Arhive, No. 03-112 (2003); www.ma.utexas.edu/mp-arc/c/03/03-112.ps.gz.
A. Delshams, V. G. Gelfreich, A. Jorba, and T. M. Seara, “Exponentially small splitting of separatrices under fast quasiperiodic forcing,” Comm. Math. Phys., 189, 35–71 (1997).
A. Delshams, P. Gutierrez, and T. M. Seara, “Exponentially small splitting for whiskered tori in Hamiltonian systems: Flow-box coordinates and upper bounds,” Mathematical Physics Preprint Archive, No. 03-134 (2003); www.ma.utexas.edu/mp-arc/c/03/03-134.ps.gz.
L. H. Eliasson, “Biasymptotic solutions of perturbed integrable Hamiltonian systems,” Bol. Soc. Brasil. Mat. (N.S.), 25, No.1, 57–76 (1994).
V. G. Gelfreich, “A proof of the exponentially small transversality of the separatrices for the standard map,” Comm. Math. Phys., 201, No.1, 155–216 (1999).
V. G. Gelfreich and V. F. Lazutkin, “Splitting of separatrices: perturbation theory and exponential smallness,” Russian Math. Surveys, 56, No.3, 499–558 (2001).
H. Koch, “A renormalization group for Hamiltonians, with applications to KAM theory,” Ergodic Theory Dynam. Systems, 19, No.2, 475–521 (1999).
V. F. Lazutkin, “Splitting of separatrices for the Chirikov standard map,” Preprint VINITI, No. 6372-84 (1984).
P. Lochak, J.-P. Marco, and D. Sauzin, “On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems,” Preprint (1999).
P. Lochak, “Effective speed of Arnold’s diffusion and small denominators,” Phys. Lett. A, 143, No.1–2, 39–42 (1990).
P. Lochak, “Canonical perturbation theory via simultaneous approximation,” Russian Math. Surveys, 47, No.6, 57–133 (1992).
J. Lopes Dias, “Renormalization scheme for vector fields on T 2 with a Diophantine frequency,” Nonlinearity, 15, No.3, 665–679 (2002).
A. Pronin and D. Treschev, “Continuous averaging in multi-frequency slow-fast systems,” Regul. Chaotic Dyn., 5, No.2, 157–170 (2000).
M. Rudnev and S. Wiggins, “On a homoclinic splitting problem,” Regul. Chaotic Dyn., 5, No.2, 227–242 (2000).
D. Sauzin, “A new method for measuring the splitting of invariant manifolds,” Ann. Sci. Ecole Norm. Sup. (4), 34, No.2, 159–221 (2001).
C. Simo, “Averaging under fast quasiperiodic forcing,” in: Hamiltonian Mechanics: Integrability and Chaotic Behavior, J. Seimenis (ed.), Plenum, New York (1994), pp. 13–34.
C. Simo and C. Valls, “A formal approximation of the splitting of separatrices in the classical Arnold’s example of diffusion with two equal parameters,” Nonlinearity, 14, No.6, 1707–1760 (2001).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 87–121
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Delshams, A., Gutierrez, P. Exponentially Small Splitting of Separatrices for Whiskered Tori in Hamiltonian Systems. J Math Sci 128, 2726–2746 (2005). https://doi.org/10.1007/s10958-005-0224-x
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DOI: https://doi.org/10.1007/s10958-005-0224-x