ON THE NONSTATIONARY PASSAGE THROUGH BIFURCATIONS IN RESONANTLY FORCED HAMILTONIAN OSCILLATORS

Many non-linear dynamical systems are characterized by bifurcation par ameters which vary with time. This paper presents an analytical framew ork to analyze the dynamics of weakly non-linear, single-degree-of-fre edom, harmonically excited Hamiltonian systems as a bifurcation parame ter varies slowly across the bifurcation points, or points of instabil ity. Formal results concerning the slowly varying normal forms of such systems in the neighborhood of a bifurcation point, are derived. The use of matched asymptotic expansions and non-linear boundary layers, i n analyzing the slowly varying normal forms is briefly summarized. The simplified boundary layer equations capture all the essential ''local '' dynamics of the original Hamiltonian system during transition acros s a bifurcation. The developed theory is illustrated through derailed analyses of the dynamics of two classical Hamiltonian systems: the for ced, undamped, non-linear Mathieu equation and the forced, undamped Du ffing equation as the excitation frequency slowly varies (non-stationa ry excitation) across the points of instabilities (simple bifurcations ) in these systems. (C) 1998 Elsevier Science Ltd. All rights reserved .