Hopf-flip bifurcation of high dimensional maps and application to vibro-impact systems

This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called ‘‘time-2τ² map’’ of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T¹ (Hopf invariant circles), tori 2T¹ and tori 2T² depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms’ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.The project supported by the National Natural Science Foundation of China (10472096)The English text was polished by Ron Marshall.