Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T06:08:57.340Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical Features in a Slow-fast Piecewise Linear HamiltonianSystem

Published online by Cambridge University Press:  17 September 2013

A. Kazakov ,
N. Kulagin  and
L. Lerman
A. Kazakov
Affiliation:
Lobachevsky State University of Nizhni Novgorod, Russia
N. Kulagin
Affiliation:
The State University of Management, Moscow, Russia
L. Lerman*
Affiliation:
Lobachevsky State University of Nizhni Novgorod, Russia
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

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

Keywords

slow-fast systemHamiltonianpiecewise linearhomoclinicperiodicnonsmooth bifurcation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 155 - 172
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alfimov, G. L., Eleonsky, V. M., Lerman, L. M.. Solitary wave solutions of nonlocal sine-Gordon equations. Chaos, v.8 (1998), No.1, 257–271. CrossRefGoogle ScholarPubMed
Amick, C. J., Kirschgässner, K.. A theory of solitary water-waves in the presence of surface tension. Arch. Ration. Mech. Anal., v.105 (1989), 1–49. CrossRefGoogle Scholar
J. Amick, C., McLeod, J. B.. A singular perturbation problem in water waves, Stab. Appl. Anal. Contin. Media. v.1 (1992), 127–148. Google Scholar
V. I. Arnold, A. G. Givental. Symplectic geometry. In the book "Encyclopaedia of Mathematical Sciences", vol. 4, Springer-Verlag, Berlin-Heidelberg-New York.
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.
M. di Bernardo, C. Budd, A. Champneys, P. Kowalzcyk. Piecewise-smooth Dynamical Systems. Theory and Applications. Springer-Verlag, New York, 2008.
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, 1881–1908. Google Scholar
Eckhaus, W.. Singular perturbations of homoclinic orbits in R 4. SIAM J. Math. Anal., v.23 (1992), 1269–1290. CrossRefGoogle Scholar
Feigin, M. I.. On the generation of sets of subharmonic modes in a piecewise continuous system. Prikl. Matem. Mekh., v.38 (1974), 810–818 (in Russian). Google Scholar
Feigin, M. I.. On the structure of C-bifurcation boundaries of piecewise continuous systems. Prikl. Matem. Mekh., v.42 (1978), 820–829 (in Russian). Google Scholar
Feigin, M. I.. The increasingly complex structure of the bifurcation tree of a piecewise-smooth system. Journal of Appl. Maths. Mech., v.59 (1995), 853–863. CrossRefGoogle Scholar
M. I. Feigin. Forced Oscillations in Systems with Discontinuous Nonlinearities. Nauka P.H., Moscow, 1994 (in Russian).
Lerman, L. and Gelfreich, V.. Slow-fast Hamiltonian Dynamics Near a Ghost Separatrix Loop. J. Math. Sci., Vol.126 (2005), No.5, 1445–1466. CrossRefGoogle Scholar
Grotta Ragazzo, C.. Nonintegrability of some Hamiltonian systems, scattering and analytic continuation. Comm. Math. Phys. v.166 (1994), No. 2, 255–277. CrossRefGoogle Scholar
Vanderbauwhede, A., Fiedler, B.. Homoclinic period blow-up in reversible and conservative system. ZAMP, v.43 (1992), 291–318. CrossRefGoogle Scholar
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, 397–408. CrossRefGoogle Scholar
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).
Simpson, D. J. W., Meiss, J. D.. Simultaneous border-collision and period-doubling bifurcations. Chaos, v.19 (2009), 033146. CrossRefGoogle Scholar
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), 95–126. CrossRefGoogle Scholar
Neishtadt, A. I.. On separation of motions in systems with rapidly rotating phases. Appl. Math. Mech., v.48 (1984), 197–204. Google Scholar
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.
Shilnikov, L. P.. On the Poincaré-Birkhoff Problem. USSR Math. Sb., v.3 (1967), 415–443. Google Scholar