Painlevé analysis, Lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type

The complex geometry of the Kowalewski-Painlevé analysis

: A van der Pol relaxation oscillator that is subjected to external sinusoidal forcing can exhibit stable and unstable periodic and almost periodic responses. For some forcing amplitudes it even happens that two stable subharmonics having different periods may coexist. The stable responses of such forced oscillators are investigated. By numerically computing the rotation number of stable oscillations for various values of the forcing amplitude and oscillator tuning, descriptions are obtained of regions of phase locking, successive bifurcation of stable subharmonic and almost periodic oscillations, and overlap regions where two distinct stable oscillations can coexist.

Frequency entrainment of a forced van der Pol oscillator

This paper deals with the nonlinear dynamics of the biological system modeled by the multi-limit cycles Van der Pol oscillator. Both the autonomous and non-autonomous cases are considered using the analytical and numerical methods. In the autonomous state, the model displays phenomenon of birhythmicity while the harmonic oscillations with their corresponding stability boundaries are tackled in the non-autonomous case. Conditions under which superharmonic, subharmonic and chaotic oscillations occur in the model are also investigated. The analytical results are validated and supplemented by the results of numerical simulations. � 2005 Elsevier Ltd. All rights reserved.

Nonlinear dynamics and strange attractors in the biological system

Using the Painleve property of partial differential equations, the auto‐Backlund transformation and Lax pairs for a Korteweg–de Vries (KdV) equation with time‐dependent coefficients are obtained. The Lax pair criterion also makes it possible for some new models of the variable coefficient KdV equation to be found that can represent nonsoliton dynamical systems. This can explain the wave breaking phenomenon in variable depth shallow water.

Auto‐Bäcklund transformation, Lax pairs, and Painlevé property of a variable coefficient Korteweg–de Vries equation. I

Various limit cycle systems in the plane with an external periodic excitation are investigated from the point of view of chaotic behavior. Numerical studies are performed and a singular point analysis is carried out.

Chaos in limit cycle systems with external periodic excitations

The Hamiltonian H(x,p)=1/2(p12)+1/2(p22)+1/2(x12x22) is investigated. The different aspects (singular point analysis, stability analysis, numerical treatment etc.) for studying chaotic behaviour are discussed. Moreover, the authors discuss the quantised version of the Hamiltonian.

Chaotic behaviour of a Hamiltonian with a quartic potential

Various aspects of the integrability of dynamical systems are discussed with the help of the singular point analysis. In particular the connection with the Painleve property is described. Several examples will serve as illustrations.

Integrability of dynamical systems and the singular-point analysis

This note is concerned with the behaviour of the dynamical system x = y, y = −x−μ sin y which is important in electrical circuit theory.

Limit cycle oscillations and a perturbed harmonic oscillator

The spectrum of the two point Hubbard model with phonon coupling is studied. In particular the connection with quantum chaos is discussed.

http://w2agz.com/Library/Superconductivity/Steeb,%20Hubbard+Phonon,%20Phys.%20Scr.%2034,%20245%20(1986).pdf

A. Kunick

Papers

Chaos in limit cycle systems with external periodic excitations

Chaotic behaviour of a Hamiltonian with a quartic potential

Integrability of dynamical systems and the singular-point analysis

Limit cycle oscillations and a perturbed harmonic oscillator

Finite Hubbard Model with Phonon Coupling