Analytic approximation of periodic orbits with renormalization group

Abstract The renormalization group analysis has been proposed to eliminate secular terms in perturbation solutions of differential equations and thus expand the domain of their validity. Here we extend the method to treat periodic orbits or limit cycles. Interesting normal forms could be derived through a generalization of the concept 'resonance', which offers nontrivial analytic approximations. Compared with traditional techniques such as multi-scale methods, the current scheme proceeds in a very straightforward and simple way, delivering not only the period and the amplitude but also the transient path to limit cycles. The method is demonstrated with several examples including Duffing oscillator, Van der Pol equation and Lorenz equation. The obtained solutions match well with numerical results as well as those derived by traditional analytic methods.