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

Amartya Sarkar, Partha Guha, Anindya Ghose-Choudhury, J. K. Bhattacharjee, A. K. Mallik, P. G.L. Leach

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Abstract

A variant of the generalized Riccati system of equations, ẍ+αẋx 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, α c = 2√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, ẍ + (2n + 3)ẋx 2n+1 + x 4n+3 + ω 2x = 0, exhibits isochronous oscillations. The correctness of the conjecture is established numerically.

Original languageBritish English
Article number415101
JournalJournal of Physics A: Mathematical and Theoretical
Volume45
Issue number41
DOIs
StatePublished - 19 Oct 2012

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