A finite atlas for solution manifolds of differential systems with discrete state-dependent delays. (Q2128414)

The paper studies differential equations with \(\mathbf{k}\) state-dependent discrete delays, all of which factor through the same linear operator \(L\) with finite dimensional range. The equations take the form \[\dot x(t)=g[x(t-d_1(Lx_t)),\dots,x(t-d_{\mathbf{k}}(Lx_t))],\] with \(x(t)\in\mathbb{R}^n\), and \(x_t\in C^0([-r,0],\mathbb{R}^n)\) denoting the segment of \(x\) at time \(t\) in the usual way. The equation can be rewritten as \(\dot x(t)=f(x_t)\) with a functional \(f\) given by \(g\). It is known that the solution manifold \(X_f\subset C^1([-h,0],\mathbb{R}^n)\) (we briefly write \(C^1\) from now on), described by the compatibility condition \(\phi'(0)=f(\phi)\), is an appropriate state space for the dynamical system generated by the equation. An atlas with at most \(2^{\mathbf{k}}\) charts for \(X_f\) is constructed, where every chart corresponds to a subset \[X_{fJ}:=\{\phi\in X_f\;|\; d_j(L\phi)=0\text{ exactly for }j\in J\},\quad\text{with }J\subset\{1,\dots,\mathbf{k}\}.\] The chart domains are `almost graphs' over parts of the linear space \(X_0=\{\phi\in C^1\;|\;\phi'(0)=0\}\), that is, parametrized by maps of the form \(\zeta\mapsto\zeta +\alpha(\zeta)\) with \(\alpha\) taking values in a \(\zeta\)-dependent direct summand of \(X_0\) in \(C^1\), and \(\alpha(\zeta)=0\) if \(\zeta\in X_f\). First, to obtain a chart for the set \(X_{f\{1,\dots,\mathbf{k}\} }\subset X_f\) where all delays vanish, functions \(\chi_\nu\) in the kernel of \(L\) with \(\chi_{\nu}(0)=0,\;\chi_{\nu}'(0=e_{\nu}\) (the \(\nu\)-th unit vector) are used. These define a projection \(R\) from \(C^1\) to \(X_0\), with the \(\chi_{\nu}\) spanning the kernel of \(R\), namely \[R(\phi):=\phi -\sum_{\nu}\phi_{\nu}'(0)\chi_\nu.\] Under additional assumptions, for example, boundedness of \(g\), it turns out that \(R\) defines a (global) chart for \(X_{f\{1,\dots,\mathbf{k}\}}\). Second, an analogous, but more involved construction for subsets \(J\neq\{1,\dots,\mathbf{k}\}\) has to use `point-dependent' functions \(\eta_{J\nu}(L\phi)\) for \(\phi\in X_{fJ}\), and the `projections' \(\displaystyle R^J(\phi):=\phi -\sum_{\nu}\phi_{\nu}'(0)\eta_{J\nu}(L\phi).\) This results in an `almost graph' representation of the sets \(X_{fJ}\), and restrictions of the maps \(R^J\) as (global) charts for \(X_{fJ}\). Since \(X_f\) is the disjoint union of all \(X_{fJ}\) (including \(J=\emptyset\)), the mentioned atlas is obtained.