Exploring elements of chaos in driven strings

Abstract

This paper presents a comprehensive analysis of nonlinear oscillations exhibited by driven strings. We begin with a linear, periodically driven, string in a nonlinear viscoelastic medium. We then show its reduction to the general Duffing oscillator model under the traveling wave assumption, and analyse several cases of the model including a special case corresponding to Ueda oscillator. In addition, we also study a string-oscillator problem involving an exponential (i.e., Toda) nonlinearity. We analyze each system with time series, phase portraits, bifurcation diagrams, Poincaré sections, and Lyapunov exponents. The nonlinear stiffness and driving play a major role in identifying regions of periodic motion and chaos in Duffing type of oscillators. The systematic approach provides an understanding of different elements of chaos in the nonlinear systems considered.