Poincaré-Bendixson theory for certain retarded functional differential equations
A system \(x'(t)=Ax_ t+B\Phi(Cx_ t)\) with \(A\varphi=\int^ 0_{- h}[da(\theta)]\varphi(\theta)\), \(C\varphi=\int^ 0_{- h}[de(\theta)]\varphi(\theta)\), is considered. Let \(a(z)=\int^ 0_{- a}e^{z\theta}da(\theta)\), \(c(z)=\int^ 0_{- h}e^{z\theta}dc(\theta)\). Assume that there exists \(\lambda>0\) such that no root of \(\text{det}(zI-a(z))=0\) has \(\text{Re} z=-\lambda\), and exactly two roots have \(\text{Re} z>-\lambda\). Define \(\mu(\lambda)=\sup\{| \chi(-\lambda-i\sigma)|,\sigma\in\mathbb{R}\}\), \(\chi(z)=c(z)(zI-a(z))^{-1}B\), \((| M|^ 2\) is the largest eigenvalue of \(M*M)\). The main result of the paper states that if the Lipschitz constant of \(\Phi\) is less than \(1/\mu(\lambda)\), then the \(\omega\)-limit set of a bounded positive semiorbit which does not contain critical points consists of a single periodic trajectory. An application to a specific system in Mathematical Biology is described. No references to results in the same area by the Russian school of Yakubovich.
scientific article; zbMATH DE number 90486