Stability on finite time interval and time-dependent bifurcation analysis of Duffing's equations (Q1970789)

Let \(f: \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n\) be continuously differentiable satisfying \(f(0,0)=0,\) let \(f_x (0,0)\) be singular. The authors are interested in studying the dynamic bifurcation problem \(dx/dt = f(x,\lambda)\) with \(\lambda = \lambda (\varepsilon t)\) where \(\varepsilon\) is a small parameter. To this purpose the concept of the stability on a finite time interval is introduced and some theorems related to that topic are formulated. They apply their results to the time-dependent Duffing equation \( \ddot{x} + k \dot{x} = \lambda (\varepsilon t)x-x^3\) to determine the delayed bifurcation behavior.