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Stability on finite time interval and time-dependent bifurcation analysis of Duffing’s equations. (English) Zbl 0947.34025

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.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34K18 Bifurcation theory of functional-differential equations
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