We study a three-dimensional dynamical system, which is obtained as a pseudo-spectral approximation to a free boundary model problem. This free-boundary model was designed to capture sailient nonlinearities that govern certain solid-combustion and phase-transition phenomena. It has been shown to develop complex dynamical patterns, including a Hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos, reverse sequences, and sequences followed by Shilnikov type trajectories. The three-dimensional dynamical system turns out to be a qualitatively accurate approximation of the free boundary problem and is capable of generating its major dynamical patterns. Results of numerical simulations demonstrate a variety of dynamical scenarios that are controlled by the boundary kinetics functions inherited from the original problem. A computer-assisted bifurcation analysis of a codimension-two bifurcation is performed. It uncovers some novel mechanisms of exchange of stability that are manifested in the dynamical system for a wide range of parameters.

This work was partly supported by the National Science Foundation through grants DMS-9502142 and DMS-9510728, and the Alfred P. Sloan Foundation through a Sloan Research Fellowship.

Back to Gregor Kovacic's Home Page