Controlling Nonlinear Dynamics in a Two-Well Impact System.

Abstract

A method for controlling nonlinear dynamics based on avoiding homoclinic intersection is systematically implemented to perform a numerical analysis of the control induced modifications of the steady attractors and bifurcation scenario of a two-well impact system. The work is divided into two parts. This paper (Part I) deals with the analysis of the harmonic (reference) and global optimal excitations, which are both symmetric. The bifurcation diagrams obtained for increasing values of the excitation amplitude show there exist a "basic" attractor and other "complementary" solutions. The range of stability of the principal complementary attractors is numerically established, and the mechanisms leading to their disapperance are identified. The role of classical and nonclassical local bifurcations in determining the system response is emphasized. Chaotic attractors are seen to appear and disappear both by classical period doubling route to chaos and by sudden changes. Subductions, boundary and interior crises are repeatedly observed. By comparison of the system response under different excitations we obtain information on the performances of global control, which furnishes relatively low gain in terms of regularization but succeeds in controlling the whole phase space.