Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity

Abstract This paper is a sequel to Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125–149.], where a system of coupled oscillators with widely separated frequencies and energy-preserving quadratic non-linearity is studied. We analyze the system for a different set of parameter values compared with those in Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125–149.]. In this set of parameters, the manifold of equilibria are non-compact. This turns out to have an interesting consequence to the dynamics. Numerically, we found interesting bifurcations and dynamics such as torus (Neimark–Sacker) bifurcation, chaos and heteroclinic-like behavior. The heteroclinic-like behavior is of particular interest since it is related to the regime behavior of the atmospheric flow which motivates the analysis in Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125–149.] and this paper.

[1]  S. Chow,et al.  A normally elliptic Hamiltonian bifurcation , 1993 .

[2]  P. Holmes,et al.  Homoclinic orbits and chaos in three– and four–dimensional flows , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  Daan Crommelin,et al.  Homoclinic Dynamics: A Scenario for Atmospheric Ultralow-Frequency Variability , 2002 .

[4]  Ali H. Nayfeh,et al.  Nonlinear interactions in a parametrically excited system with widely spaced frequencies , 1995 .

[5]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[6]  Ferdinand Verhulst,et al.  Hamiltonian systems with widely separated frequencies , 2003 .

[7]  Interactions of Andronov-Hopf and Bogdanov-Takens bifurcations , 1998, math/9811178.

[8]  M. Nicol.  CHAOS NEAR RESONANCE (Applied Mathematical Sciences 138) By G. HALLER: 427 pp., £53.50, ISBN 0-387-98697-9 (Springer, New York, 1999). , 2001 .

[9]  G. Haller,et al.  Chaos near resonance , 1999 .

[10]  J. M. Tuwankotta.  Widely separated frequencies in coupled oscillators with energy-preserving quadratic nonlinearity , 2003 .

[11]  Carles Simó,et al.  Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing , 2002 .

[12]  Daan Crommelin,et al.  Regime Transitions and Heteroclinic Connections in a Barotropic Atmosphere , 2003 .

[13]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[14]  R. Vitolo.  Bifurcations of attractors in 3D diffeomorphisms : a study in experimental mathematics , 2003 .

[15]  Ali H. Nayfeh,et al.  NONLINEAR INTERACTIONS BETWEEN TWO WIDELY SPACED MODES — EXTERNAL EXCITATION , 1993 .