Homoclinic Transition to Chaos in the Duffing Oscillator Driven by Periodic Piecewise Linear Forces

Abstract

We have applied the Melnikov criterion to examine a homoclinic bifurcation and transition to chaos in the Duffing oscillator driven by different forms of periodic piecewise linear forces. The periodic piecewise linear forces considered are triangular, hat, trapezium, quadratic and rectangular types of forces. For all the forces, an analytical threshold condition for the homoclinic transition to chaos is derived using Melnikov method and Melnikov threshold curves are drawn in a parameter space. Using the Melnikov threshold curves, we have found a critical forcing amplitude fc above which the system may behave chaotically. We have analyzed both analytically and numerically the homoclinic transition to chaos in the Duffing system with เธฎเธ•-parametric force also. The predictions from Melnikov method have been further verified numerically by integrating the governing equation and finding areas of chaotic behaviour.