On the application of the homotopy analysis method to limit cycles' approximation in planar self-excited systems (Q450493)

The paper deals with the large amplitude limit cycles and their frequency of the planar autonomous systems NEWLINE\[NEWLINE\dot{x}=P(x,y,\lambda),\;\dot{y}=Q(x,y,\lambda),NEWLINE\]NEWLINE where \(P(x,y,\lambda)\) and \(Q(x,y,\lambda)\) are analytic functions of theirs variables, \(x\) and \(y\) are the phase variables and \(\lambda\) is the control parameter. By applying the homotopy analysis method proposed by \textit{S. Liao} [Int. J. Numer. Methods Fluids 14, No. 10, 1173--1191 (1992; Zbl 0754.76065)] the authors deduce analytical approximations of limit cycles and their frequencies in mentioned self-excited systems with strong nonlinearity. As an example, a Rosenzweig-MacArthur predator-prey model is studied in details. The high accuracy of the analytical results are illustrated by comparing with those of numerical integrations.