On the application of the homotopy analysis method to limit cycles’ approximation in planar self-excited systems

In this paper, the homotopy analysis method is applied to deduce analytical approximations of limit cycles and their frequencies in general planar self-excited systems with strong nonlinearity. After changing general planar self-excited systems to the canonical forms by several linear transformations, the auxiliary linear operators and the initial guess of solutions are introduced. Hence, the homotopy analysis solving is set up. Importantly, in solving the higher-order deformation equations, the idea of a perturbation procedure of limit cycles’ approximation proposed in the setting of second-order self-excited equations is embedded. As an application, a Rosenzweig–MacArthur predator–prey model is studied in detail. By choosing the suitable convergence-control parameters, the accurately analytical approximations of the large amplitude limit cycles and their frequency of the model are obtained. The high accuracy of the analytical results are illustrated by comparing with those of numerical integrations.

► Analytical approximations of limit cycles and their frequencies in general planar self-excited systems are obtained by using the homotopy analysis method.
► The generality of the homotopy analysis method is shown.
► A predator–prey model is solved by using the present homotopy analysis procedure.