Global and Local Control of homoclinic and heteroclinic bifurcations

A comprehensive resonant optimal control method is developed and discussed for suppressing homoclinic and heteroclinic bifurcations of a general one-degree-of-freedom nonlinear oscillator. Based on an adjustable phase shift, the primary resonant optimal control method is presented. By solving an optimization problem for the optimal amplitude coefficients to be used as the control parameters, the force term as the controller can be designed. Three kinds of resonant optimal control methods are compared. The control mechanism of the primary resonant optimal control method is to enlarge to the largest possible degree the control region where homoclinic and/or heteroclinic transversal intersections do not occur, and this is accomplished at lowest cost. It is shown that the primary resonant optimal control method has much better performance than the superharmonic resonant optimal control method, and it works well even when the superharmonic optimal control method fails. In particular, one new global optimal control method is presented, whose central idea is to find a frequency such that the asymmetric homoclinic bifurcations or the multiple homoclinic and heteroclinic bifurcations can attain the same critical values. On the basis of these same critical bifurcation values, chaos resulting from asymmetric homoclinic or multiple homoclinic and heteroclinic bifurcations can be effectively suppressed by the primary resonant optimal control method. This is confirmed by two illustrative examples. The theoretical analyses concerning the suppression of local and global homoclinic and heteroclinic bifurcations are in agreement with the numerical simulations, including the identification of the stable and unstable manifolds and the basins of attraction.