Effects of periodic forcing on delayed bifurcations

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

$$\begin{gathered} \frac{{\partial u}}{{\partial t}} = f\left( {u, I_i + \varepsilon t} \right) + \varepsilon g\left( {u, I_i + \varepsilon t, \varepsilon , t} \right) \hfill \\ \left. {u\left( t \right)} \right|_{t = 0} = u_0 \left( {I_i } \right) + O\left( \varepsilon \right) \hfill \\ \end{gathered} $$

((0.1))

whereu ₀(I) is the solution off(u ₀(I), I)=0 andI(t)=I _i+εt is a slowly varying parameter that moves past a critical pointI_ of the system so that the linear stability aroundu ₀(I) changes from stable to unstable atI_. General results are given with respect to the effects of the perturbation εg(u,I(t),ε, t) to several important types of dynamical systems

$$\frac{{\partial u}}{{\partial t}} = f\left( {u,I_i + \varepsilon t} \right)$$

((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 ofg(…,…,…,t) ont is a constantΩ which is not a resonant frequency; (2) in case the frequency ofg(…,…,…,t) ont isΩ=Ω(I _i+εt) that is slowly varying, the resonance frequencies where the delayed bifurcations might be destructed are shifted downward or upward depending onΩ′(I_)>0 orΩ′(I_)<0; and (3) delayed pitchfork (simple eigenvalue) bifurcations occur in a codimension one parameter family of periodic perturbations. (1) is a rigorous analysis of the results in [3], (2) is a new and interesting phenomenon, and (3) is a generalization of the results of Diener [8] and Schecter [19].