MELNIKOV'S METHOD FOR NON-LINEAR OSCILLATORS WITH NON-LINEAR EXCITATIONS

✍ Scribed by J. Garcia-Margallo; J.D. Bejarano

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 189 KB
• Volume: 212
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1997.1443

✦ Synopsis

The response of a non-linear oscillator of the form x¨+ f(A, B, x) = og (E, m, w, k, t), where f(A, B, x) is an odd non-linearity and o is small, for A Q 0 and B q 0 is considered. The homoclinic orbits for the unperturbed system are obtained by using Jacobian elliptic functions with the generalized harmonic balance method. Also the chaotic limits of this equation are studied with a generalized Melnikov function, M 0 (E, m, x˙, w, k, t0), depending on the variable k. A function R 0 (E, m, w, k) is defined such that there only exists chaotic motion if E/m q R 0 with k from 0•51 to 0•99. It is demonstrated with Poincare´maps in the phase plane that there is good agreement between these predictions and the numerical simulations of the Duffing-Holmes oscillator using the fourth-order Runge-Kutta method of numerical integration.