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Dynamical Features in a Slow-fast Piecewise Linear Hamiltonian System

  • A. Kazakov (a1), N. Kulagin (a2) and L. Lerman (a1)


We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equilibrium of the saddle-center type can have a sequence of small parameter values for which a one-round homoclinic orbit to this equilibrium exists. This contrasts with the well-known findings by Amick and McLeod and others that solutions of such type do not exist in analytic Hamiltonian systems, and that the separatrices are split by the exponentially small quantity. We also discuss existence of homoclinic trajectories to small periodic orbits of the Lyapunov family as well as symmetric periodic orbits near the homoclinic connection. Our further result, illustrated by simulations, concerns the complicated structure of orbits related to passage through a non-smooth bifurcation of a periodic orbit.


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[1] Alfimov, G. L., Eleonsky, V. M., Lerman, L. M.. Solitary wave solutions of nonlocal sine-Gordon equations. Chaos, v.8 (1998), No.1, 257271.
[2] Amick, C. J., Kirschgässner, K.. A theory of solitary water-waves in the presence of surface tension. Arch. Ration. Mech. Anal., v.105 (1989), 149.
[3] J. Amick, C., McLeod, J. B.. A singular perturbation problem in water waves, Stab. Appl. Anal. Contin. Media. v.1 (1992), 127148.
[4] V. I. Arnold, A. G. Givental. Symplectic geometry. In the book "Encyclopaedia of Mathematical Sciences", vol. 4, Springer-Verlag, Berlin-Heidelberg-New York.
[5] V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt. Mathematical aspects of classical and celestial mechanics. Encycl. Math. Sci., 3, Springer-Verlag, New York-Berlin, 1993.
[6] M. di Bernardo, C. Budd, A. Champneys, P. Kowalzcyk. Piecewise-smooth Dynamical Systems. Theory and Applications. Springer-Verlag, New York, 2008.
[7] di Bernardo, M., Feigin, M., Hogan, S.J., Homer, M.E.. Local Analysis of C-Bifurcations in n-Dimensional Piecewise Smooth Dynamical Systems. Chaos, Solitons & Fractals, v.10 (1999), No.11, 18811908.
[8] Eckhaus, W.. Singular perturbations of homoclinic orbits in R 4. SIAM J. Math. Anal., v.23 (1992), 12691290.
[9] Feigin, M. I.. On the generation of sets of subharmonic modes in a piecewise continuous system. Prikl. Matem. Mekh., v.38 (1974), 810818 (in Russian).
[10] Feigin, M. I.. On the structure of C-bifurcation boundaries of piecewise continuous systems. Prikl. Matem. Mekh., v.42 (1978), 820829 (in Russian).
[11] Feigin, M. I.. The increasingly complex structure of the bifurcation tree of a piecewise-smooth system. Journal of Appl. Maths. Mech., v.59 (1995), 853863.
[12] M. I. Feigin. Forced Oscillations in Systems with Discontinuous Nonlinearities. Nauka P.H., Moscow, 1994 (in Russian).
[13] Lerman, L. and Gelfreich, V.. Slow-fast Hamiltonian Dynamics Near a Ghost Separatrix Loop. J. Math. Sci., Vol.126 (2005), No.5, 14451466.
[14] Grotta Ragazzo, C.. Nonintegrability of some Hamiltonian systems, scattering and analytic continuation. Comm. Math. Phys. v.166 (1994), No. 2, 255277.
[15] Vanderbauwhede, A., Fiedler, B.. Homoclinic period blow-up in reversible and conservative system. ZAMP, v.43 (1992), 291318.
[16] Koltsova, O. Yu., Lerman, L. M.. Periodic and homoclinic orbits in a two-parameter unfolding of a Hamiltonian system with a homoclinic orbit to a saddle-center. Int. J. Bifurcation & Chaos. v.5 (1995), No.2, 397408.
[17] L. M. Lerman. Hamiltonian systems with loops of a separatrix of a saddle-center. in "Methods of the Qualitative Theory of Differential Equations", Gor’kov. Gos. Univ., Gorki, 1987, 89–103 (in Russian); Selecta Math. Soviet., v.10 (1991), 297–306 (in English).
[18] Simpson, D. J. W., Meiss, J. D.. Simultaneous border-collision and period-doubling bifurcations. Chaos, v.19 (2009), 033146.
[19] Mielke, A., Holmes, P., O’Reilly, O.. Cascades of homoclinic orbits to, and chaos near a Hamiltonian saddle-center. J. Dyn. Different. Equat., v.4 (1992), 95126.
[20] Neishtadt, A. I.. On separation of motions in systems with rapidly rotating phases. Appl. Math. Mech., v.48 (1984), 197204.
[21] S. Smale. Diffeomorphisms with infinitely many periodic points. in "Differential and Combinatorial Topology," Ed. S. Cairns. Princeton Math. Ser., Princeton, NJ: Princeton Univ. Press, 63–80.
[22] Shilnikov, L. P.. On the Poincaré-Birkhoff Problem. USSR Math. Sb., v.3 (1967), 415443.


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Dynamical Features in a Slow-fast Piecewise Linear Hamiltonian System

  • A. Kazakov (a1), N. Kulagin (a2) and L. Lerman (a1)


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