Effects of periodic forcing on delayed bifurcations
Jianzhong Su1997, Journal of Dynamics and Differential Equations
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Abstract
This work is concerned with the rigorous analysis of the effects of small periodic forcing (perturbations) on the dynamical systems which present some interesting phenomena known as delayed bifurcations. We study the dynamical behavior of the system ~u "r f(u, Ii + et) + sg(u, li + st, 8, t) Of u(t)l,.o ffi uo(li) + 0(8) (0.1) where uo(l) is the solution off(uo(l), I) •O and l(t)=-li+8t is a slowly varying parameter that moves past a critical point I_ of the system so that the linear stability around uo(l) changes from stable to unstable at I_. General results are given with respect to the effects of the perturbation 8g(u, I(t), e, t) to several important types of dynamical systems ~u =f(u, lj +80 (0.2) which present dynamical patterns that there exist persistent unstable solutions in the dynamical systems (delayed bifurcations) in contrast to bifurcations in the classical sense. It is shown that (1) the delayed bifurcations persist if the frequency of g(.,.,., t) on f is a constant co which is not a resonant frequency; (2) in case the frequency of g(.,.,.,t) on t is co =. co(lt + st) that is slowly varying, the resonance frequencies where the delayed bifurcations might be destructed are shifted downward or upward depending on co'(l_)>O or co'(I_) <0; and (3) delayed pitchfork (simple eigenvalue) bifurcations occur in
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