Delayed loss of stability and excitation of oscillations in nonautonomous differential equations with retarded argument
- Lani-Wayda, Bernhard
- Schneider, Klaus R.
2010 Mathematics Subject Classification
- 34K12 34K06
- Nonautonomous delay equations, slowly changing parameters, delayed loss of stability, excitation of oscillations
Assume that zero is a stable equilibrium of an ODE ẋ = ƒ(𝑥, λ) for parameter values λ < λ0, and becomes unstable for λ > λ0. If we suppose that λ(t) varies slowly with t, then, under some conditions, the trajectories of the nonautonomous ODE ẋ = ƒ(𝑥, λ (t)) stay close to zero even long after λ(t) has crossed the value λ0. This phenomenon is called νdelayed loss of stabilityν and is well-known for ODEs. In this paper, we describe an analogous phenomenon for delay equations of the form ẋ(t) = ƒ(t, 𝑥 (t-1)). Further, we point out a difference between delay equations and ODEs: The inhomogeneity 𝒽 in the linear equation ẋ (t) = c𝑥 (t-1) + 𝒽(t) inevitably leads to an excitation of the most unstable modes of oscillation of the homogeneous equation, even if all segments 𝒽t are contained in a space of more rapidly decaying solutions for the homogeneous equation.