Phase asymptotic semiflows, Poincaré's condition, and the existence of stable limit cycles (Q1906171)

The paper deals with phase asymptotic semiflows. The authors prove that any stable orbit at which the semiflow is phase asymptotic limits to a stable periodic orbit. The basic results (Poincaré-Bendixson-type theorems) are introduced in Section 4 of the paper. Let \(\varphi= \varphi (t)\), \(t\in {\mathbb{R}}\) be the solution of the differential equation \(\dot x= f(x)\), where \(f= f(x)\) is a \(C^1\)-smooth function, \(\Gamma_+= \{\varphi (t): t\geq 0\}\), \(\Omega\) is the \(\omega\)-limit set of \(\Gamma_+\), and let the differential equation \[ \dot z= {{\partial f^2} \over {\partial x}} (\varphi (t))z \] be uniformly asymptotically stable, where \( {{\partial f^2} \over {\partial x}} (x)\) is the second additive compound of \({{\partial f} \over {\partial x}} (x)\). Then if \(\Omega\) contains no equilibrium, then it is a periodic orbit; if \(\Omega\) contains an equilibrium \(x_0\), then either \(x_0\) is asymptotically stable or it has an \((n-1)\)-dimensional stable manifold together with a 1-dimensional unstable manifold; if \(\Omega\) contains an equilibrium \(x_0\) and the differential equation \(\dot y= {{\partial f} \over {\partial x}} y\) is uniformly stable, then \(\Omega= \{x_0\}\).