Chaotic Behavior of the Biharmonic Dynamics System † Vladimir

Motion of a biharmonic system under action of small periodic force and small damped force is studied. The biharmonic oscillator is a physical system acting under a biharmonic force like: θ θ 2 sin sin b a + . The article contains biharmonic oscillator analysis, phase space research, and analytic solutions for separatrixes. The biharmonic oscillator performs chaotic motion near separatrixes under small perturbations. Melnikov method gives analytical criterion for heteroclinic chaos in terms of system parameters. A transition from chaotic to regular motion of the biharmonic oscillator was found as the heteroclinic chaos can be removed by increasing the coeffi cient of a damping force. The analytical results obtained using Melnikov method has been confi rmed by a good match with numeric research. Introduction Reduction of the Biharmonic Dynamics System to the Duffi ng Equation As is well-Known, a nonlinear system can perform a chaotic motion under the action of periodic forces [1–4]. Frequently the Duffi ng equation is used to illustrate chaos [1–4], and the chaotic behavior of various forms of the Duffi ng equation [5], some of which exhibit two-frequency excitation [6] as well as the chaotic motion of Duffi ng system with bounded noise [7] have been investigated. However the Duffi ng equation in the expanded (generalized) form has many mechanical applications and it can be interesting to researchers. Chaotic Behavior of the Biharmonic Dynamics System 627