Local invariant manifolds for delay differential equations with state space in $C^1((-\infty,0],\mathbb{R}^n)$

Abstract

Consider the delay differential equation $x'(t)=f(x_t)$ with the history $x_t:(-\infty,0]\to\mathbb{R}^n$ of $x$ at 'time' $t$ defined by $x_t(s)=x(t+s)$. In order not to lose any possible entire solution of any example we work in the Fréchet space $C^1((-\infty,0],\mathbb{R}^n)$, with the topology of uniform convergence of maps and their derivatives on compact sets. A previously obtained result, designed for the application to examples with unbounded state-dependent delay, says that for maps $f$ which are slightly better than continuously differentiable the delay differential equation defines a continuous semiflow on a continuously differentiable submanifold $X\subset C^1$ of codimension $n$, with all time-t-maps continuously differentiable. Here continuously differentiable for maps in Fréchet spaces is understood in the sense of Michal and Bastiani. It implies that $f$ is of locally bounded delay in a certain sense. Using this property - and a related further mild smoothness hypothesis on $f$ - we construct stable, unstable, and center manifolds of the semiflow at stationary points, by means of transversality and embeddings.

Keywords

• delay differential equation
• state-dependent delay
• unbounded delay
• fréchet space
• local invariant manifold