A Hartman--Grobman theorem for a class of retarded functional differential equations
The author proves the following local stability theorem for a class of retarded functional differential equations: Let \(f\) be a retarded functional differential equation on \(\mathbb{R}^ n\) with a compact attractor \(A(f)\). Let \(\varphi_ 0\) be a hyperbolic equilibrium solution of \(f\). Let \(\Phi\) be the solution map of \(f\) and let \(L\) be the solution map of its linear variational equation. Suppose that \(\Phi\) is one-to-one on \(A(f)\). Then there exists an open neighbourhood \(W\) of \(\varphi_ 0\) in \(A(f)\), an \(\varepsilon>0\) and a map \(h\) from \(A(f)\) into \(C^ 0([- r,0],\mathbb{R}^ n)\), \(r>0\), such that \(h| W\) is a homeomorphism from \(W\) into \(h(W)\), \(h(\varphi_ 0)=0\) and \(L(t)\circ h=h\circ\Phi(t)\) on \(W\), for all \(t\in (-\varepsilon,\varepsilon)\).
retarded functional differential equations
scientific article; zbMATH DE number 399059