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Session S12 - Delay and functional differential equations and applications

Tuesday, July 13, 11:40 ~ 12:15 UTC-3

On solution manifolds

Hans-Otto Walther

Justus-Liebig-Universitaet Giessen, Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Differential equations with state-dependent delays which generalize the scalar example \[ x'(t)=g(x(t),x(t-d(x_t))) \] where $g:\mathbb{R}^2\to\mathbb{R}$ and $d:C([-r,0],\mathbb{R})\to[0,r]$ are continuously differentiable, and with $x_t:[-r,0]\to\mathbb{R}$ given by $x_t(s)=x(t+s)$, define semiflows of differentiable solution operators on an associated submanifold of the state space $C^1=C^1([-r,0],\mathbb{R}^n)$. When written in the general form \[ x'(t)=f(x_t) \] with a map $f:C^1\supset U\to\mathbb{R}^n$ then the associated manifold is \[ X_f=\{\phi\in U:\phi'(0)=f(\phi)\}. \] We obtain results on the nature of $X_f$.

1. If all delays in the system are bounded away from zero then a projection $C^1\to C^1$ onto the subspace \[ H=\{\phi\in C^1:\phi'(0)=0\}=X_0 \] defines a diffeomorphism of $X_f$ onto an open subset of $H$. In other words, $X_f$ is a graph over $H$.

2. There exist $g$ and $d$ with $d(\phi)>0$ everywhere and $\inf\,d=0$ so that the manifold $X_f$ associated with the scalar example above does not admit any graph representation.

3. If all delays in the system are strictly positive (but not necessarily bounded away from zero) then $X_f$ has an ''almost graph representation`` which implies that it is diffeomorphic to an open subset of $H$.

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