Linearized instability for differential equations with dependence on the past derivative

Abstract

We provide a criterion for instability of equilibria of equations in the form $\dot x(t) = g(x_t', x_t)$, which includes neutral delay equations with state-dependent delay. The criterion is based on a lower bound $\Delta >0$ for the delay in the neutral terms, on regularity assumptions of the functions in the equation, and on spectral assumptions on a semigroup used for approximation. The spectral conditions can be verified studying the associated characteristic equation. Estimates in the $C^1$-norm, a manifold containing the state space $X_2$ of the equation and another manifold contained in $X_2$, and an invariant cone method are used for the proof. We also give mostly self-contained proofs for the necessary prerequisites from the constant delay case, and conclude with an application to a mechanical example.

Keywords

• neutral delay equations
• dependence on past derivative
• state-dependent delay
• linearized instability