Limit sets of dynamical systems (Q1874811)

The following result is proved: Assume that a system of autonomous ordinary differential equations \(dX/dt=f(x)\) is defined for all \(x\in \mathbb{R}^n\) with continuous function \(f\) and has two limit cycles \(M_\omega\) and \(M_\alpha\), which are Lyapunov asymptotically stable as \(t\to\infty\) and \(t\to -\infty\), respectively. The set of points, asymptotic as \(t\to\infty\) to \(M_\omega\) and as \(t\to -\infty\) to \(M_\alpha\) is invariant, connected and open.