Solution manifolds which are almost graphs
The solution manifold \(X_f := \{\phi \in C^1 \mid \phi'(0) = f(\varphi)\}\), where \(C^1 = C^1([-h,0], \mathbb{R}^n)\), is studied for a class of functional differential equations \(\dot{x}(t) = f(x_t)\). This class includes systems with state-dependent delays of the form \[ x_{\nu}'(t) = g_{\nu}(x(t), x_1(t-\delta_1(Lx_t)), \dots, x_n(t - \delta_n(Lx_t)),\;\nu = 1, \dots, n \] with delay functions \(\delta_{\nu}\) and a linear operator \(L\) with finite-dimensional range \(F\). In case these delays are uniformly bounded away from zero, it is shown that \(X_f\) is globally a graph over its projection onto the space \(H = \{ \phi \in C^1\, |\, \phi'(0) = 0\}\) along a suitable direct summand \(Q\) of \(H\). Complementary, an example is given where a delay can be arbitrarily close to zero and \(X_f\) admits no graph representation. If the delays are only positive but can approach zero then \(X_f\) is an `almost graph' over a subset \(R(X_f)\) of \(H\), i.e., can be written as \(\{\chi + A(\chi)\mid \chi\in R(X_f) \}\), where \(A\) a map with values \(A(\chi) \) in a \(\chi\)-dependent direct summand of \(H\). In the case where the delays have a uniform lower bound \(|z| >0\), functions \(\rho_{\nu}\) which are zero on \([-r, z] \cup\{0\}\) and satisfy \(\rho_{\nu} '(0) = e_{\nu}\) (the \(\nu\)-th unit vector) provide a suitable direct summand of \(H\). A global graph representaton requires all tangent spaces \(T_{\phi}X_f\) to be graphs over one space, and it is shown that this property does not hold in the example where the delay can approach zero. By a construction which adapts functions analogous to the \(\rho_{\nu}\) above to the values of the delays on different subsets of their domain \(W\), a point-dependent family of decompositions \(C^1([-h,0], \mathbb{R}^n) = H \oplus \mathcal{E}(w)\cdot \mathbb{R}^n\) (\(w \in W\)) with a \(C^1\) map \(\mathcal{E}:W \to \mathbb{R}^{n\times n}\) is obtained. The map \(R: \phi \mapsto \phi - \mathcal{E}(L\phi)\cdot \phi'(0)\) then defines a diffeomorphism from \(X_f\) to \(R(X_f) \subset H\), the inverse of which is the almost-graph representation. The same map \(R\) allows also to obtain a partition of \(X_f\) into shifted graphs, if one considers the sets \(\chi + (H \cap \mathrm{kern}(L))\), \(\chi \in R(X_f)\).
scientific article; zbMATH DE number 7379507
scientific article