Journal ofSound and Vibrarion (1987) 117(2), 219-232 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ AN EXAMINATION OF INITIAL CONDITION M A PS FO R THE SINUSOIDALLY EXCITED BUCKLED BEA M M O D ELED BY THE DUFFING’S EQUATION C. PEZESHKI AND E. H. DOWELL School of Engineering, Duke University, Durham, North Carolina 27706, U.S.A. (Received 26 March 1986, and in revised form 8 October 1986) The initial condition problem is studied for the dynamics of the buckled beam by using the forced Holmes-Duffing’s equation, in an attempt to understand the route to chaos. Earlier results generated by Moon [l] for different parameter values are reconsidered, and new techniques are used in order to understand more of the global behavior of the system. In particular, fractal basin boundaries are determined for two types of basins of attraction. These are correlated with previous results to help provide an explanation for observed chaotic system behavior. 1. INTRODUCTION, BACKGROUND AND MOTIVATION In this paper, the modified Holmes-Duffing equation is studied: i+rA-$A(l-A*)=F,,sinwt. (I) This is the particular form of the Duffing’s equation first studied by Holmes and Moon. It is known that this equation has solutions that are chaotic in nature for certain parameter and initial conditions. Further, it is one of the few equations for which extensive theoretical and experimental work has been performed. We examine the behavior of the system by looking at the problem through the initial condition space in the phase plane. Different types of initial condition maps are generated to ascertain differences in transient decay times for different trajectories and differences in final steady states. Fractal properties of these maps are explored in order to determine a route to chaos for the system. In a previous paper by Dowel1 and Pezeshki [2], a theoretical chaos boundary for the Duffing’s equation was obtained by using Runge-Kutta time integration for one initial conditi0.n corresponding to the position of the buckled beam at rest. This nominal chaos boundary in the force-amplitude/frequency parameter space with damping held constant was shown to agree well with an experimental (physical) chaos boundary obtained by Moon using a magnetically buckled beam. Figure 1 shows a plot of the chaotic regions that were mapped. Lines connecting the points were drawn only if the limit cycle attractors were roughly the same shape at the entrance or exit to the chaotic region. If, for example, for w = 1.0 and w = 1.1, the limit cycles existing below the chaotic regime looked the same, then a line was drawn between the points in the parameter space signifying where chaos was first observed. Generally, simulations were performed to find the first entrance and exit from chaos in the parameter space; no simulations were performed above the circles with downward arrows. Holmes [3], using a subset of Melnikov theory, derived a necessary boundary for chaos based on the existence of homoclinic orbits in the Poincare map: i.e., when F,> F,, and F< = (y&37rw) cash (rw/&). (2) 219 0022-460X/87/170219+ 14 $03.00/O @ 1987 Academic Press Limited