A local unstable manifold for differential equations with state-dependent delay
Tibor Krisztin
Discrete & Continuous Dynamical Systems - A 2003, 9(4): 993-1028 doi: 10.3934/dcds.2003.9.993
Existence and $C^N$-smoothness of a local unstable manifold at $0$ are shown for the delay differential equation $\dot x(t)=F(x_t)$ with $F:C([-h,0],\mathbb R^n)\to \mathbb R^n$, $h>0$, $F(0)=0$, under the hypotheses: There exist a linear continuous $L$ and a continuous $g$ with $F=L+g$; $0$ is a hyperbolic equilibrium of $\dot y(t)=Ly_t$; the restriction $g|_{C^k([-h,0],\mathbb R^n)}:C^k([-h,0],\mathbb R^n)\to \mathbb R^n$ is $C^k$-smooth for each $k\in$ {$1,\ldots,N$}; $D(g|_{C^1([-h,0],\mathbb R^n)})(0)=0$; in addition, for the derivatives $D^k(g|_{C^k([-h,0],\mathbb R^n)}) $, $k\in${$1,\ldots,N$}, certain extension properties hold. The conditions on $F$ are motivated and are satisfied by a wide class of differential equations with state-dependent delay.
keywords: variation-of-constans formula Functional differential equation substitution operator. state-dependent delay Lyapunov–Perron method unstable manifold
The unstable set of zero and the global attractor for delayed monotone positive feedback
Tibor Krisztin
Conference Publications 2001, 2001(Special): 229-240 doi: 10.3934/proc.2001.2001.229
Please refer to Full Text.
keywords: Delay differential equation unstable set global attractor discrete Lyapunov functional.

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