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

A variant of the generalized Riccati system of equations, , is considered. It is shown that for ? = 2n + 3 the system admits a bilagrangian description and the dynamics has a node at the origin, whereas for ? 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, , 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, , exhibits isochronous oscillations. The correctness of the conjecture is established numerically.

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