A local unstable manifold for differential equations with state-dependent delay
Tibor Krisztin
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
Please refer to Full Text.
keywords: Delay differential equation unstable set global attractor discrete Lyapunov functional.
Global stability of a price model with multiple delays
Ábel Garab Veronika Kovács Tibor Krisztin
Consider the delay differential equation \begin{equation*} \dot{x}(t)=a \Bigg(\sum_{i=1}^n b_i\big[x(t-s_i)- x(t-r_i)\big]\Bigg)-g(x(t)), \end{equation*} where $a>0$, $b_i>0$ and $0\leq s_i < r_i$ $(i\in \{1,\dots,n\})$ are parameters, $g\colon \mathbb{R} \to \mathbb{R}$ is an odd $C^1$ function with $g'(0)=0$, the map $(0,\infty)\ni \xi \mapsto g(\xi)/\xi\in\mathbb{R}$ is strictly increasing and $\sup_{\xi>0} g(\xi)/\xi>2a$. This equation can be interpreted as a price model, where $x(t)$ represents the price of an asset (e.g. price of share or commodity, currency exchange rate etc.) at time $t$. The first term on the right-hand side represents the positive response for the recent tendencies of the price and $-g(x(t))$ is responsible for the instantaneous negative feedback to the deviation from the equilibrium price.
    We study the local and global stability of the unique, non-hyperbolic equilibrium point. The main result gives a sufficient condition for global asymptotic stability of the equilibrium. The region of attractivity is also estimated in case of local asymptotic stability.
keywords: multiple delay infinite delay price model Delay differential equation stable $D$ operator. global stability neutral equation

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