PROPERTIES OF CROSS-WELL CHAOS IN AN IMPACTING SYSTEM

In this paper we present results on chaotic motions in a periodically forced impacting system which is analogous to the version of Duffing's equation with negative linear stiffness. Our focus is on the predicti on and manipulation of the cross-well chaos in this system. First, we develop a general method for determining parameter conditions under wh ich homoclinic tangles exist, which is a necessary condition for cross -well chaos to occur. We then show how one may manipulate higher harmo nics of the excitation in order to affect the range of excitation ampl itudes over which fractal basin boundaries between the two potential w ells exist. We also experimentally investigate the threshold for cross -well chaos and compare the results with the theoretical results. Seco nd, we consider the rate at which the system crosses from one potentia l well to the other during a chaotic motion and relate this to the rat e of phase space flux in a Poincare map defined in terms of impact par ameters Results from simulations and experiments are compared with a s imple theory based on phase space transport ideas, and a predictive sc heme for estimating the rate of crossings under different parameter co nditions is presented. The main conclusions of the paper are the follo wing: (1) higher harmonics can be used with some effectiveness to exte nd the region of deterministic basin boundaries (in terms of the ampli tude of excitation) but their effect on steady-state chaos is unreliab le; (2) the rate at which the system executes cross-well excursions is related in a direct manner to the rate of phase space flux of the sys tem as measured by the area of a turnstile lobe in the Poincare map. T hese results indicate some of the ways in which the chaotic properties of this system, and possibly others such as Duffing's equation, are i nfluenced by various system and input parameters. The main tools of an alysis are a special version of Melnikov's method, adapted for this pi ecewise-linear system, and ideas of phase space transport.