On the properties of a variant of the Riccati system of equations

Abstract

A variant of the generalized Riccati system of equations, $\ddot{x} + \alpha \dot{x} x^{2n+1} + x^{4n+3} = 0$, is considered. It is shown that for α = 2n + 3 the system admits a bilagrangian description and the dynamics has a node at the origin, whereas for α much smaller than a critical value the dynamics is periodic, the origin being a centre. It is found that the solution changes from being periodic to aperiodic at a critical point, $\alpha _c = 2\sqrt{2(n+1)}$, which is independent of the initial conditions. This behaviour is explained by finding a scaling argument via which the phase trajectories corresponding to different initial conditions collapse onto a single universal orbit. Numerical evidence for the transition is shown. Further, using a perturbative renormalization group argument, it is conjectured that the oscillator, $\ddot{x} + (2n+3)\dot{x}x^{2n+1}+x^{4n+3} +\omega ^2 x = 0$, exhibits isochronous oscillations. The correctness of the conjecture is established numerically.