Renormalisation group and isochronous oscillations

Abstract

We show how the condition of isochronicity can be studied for two-dimensional systems in the renormalisation group (RG) context. We find a necessary condition for the isochronicity of the Cherkas system and another class of cubic system. While the necessary condition involves a relation among the coefficients of the cubic terms, for certain choices of parameters, we can explicitly predict the specific values of the parameters for isochrony. Our conditions are satisfied by all the cases studied recently by Ghose Choudhury and Guha [J. Phys. A: Math. Theor. 43, 125202 (2010)] and Bardet et al. [Bull. Sci. Math. 135, 230 (2011)]. Finally we use our RG approach to identify a family of nonlinear isochronous oscillators not reported earlier in the literature.