Lyapunov function method for the analysis of dissipative autonomous dynamic processes (Q843686)

The paper is devoted to study the limiting behavior of autonomous dynamic processes (or semi-dynamical systems) on a compact set by means of three types of special Lyapunov functions. The authors generalize and refine the results obtained by \textit{J. Hofbauer} [Monatsh. Math. 91, 233--240 (1981; Zbl 0449.34039)] and \textit{V. Hutson} [Math. Biosci. 68, 233--246 (1984; Zbl 0534.92027)] for systems of ordinary differential equations whose right-hand sides have the structure known as a hypercycle. Sufficient conditions for the dissipativity of an autonomous dynamic process on a compact Euclidean space are presented. The results can be used in the qualitative analysis of systems of population dynamics and chemical kinetics.