TY - JOUR

T1 - On the properties of a variant of the Riccati system of equations

AU - Sarkar, Amartya

AU - Guha, Partha

AU - Ghose-Choudhury, Anindya

AU - Bhattacharjee, J. K.

AU - Mallik, A. K.

AU - Leach, P. G.L.

PY - 2012/10/19

Y1 - 2012/10/19

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84867262719&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/45/41/415101

DO - 10.1088/1751-8113/45/41/415101

M3 - Article

AN - SCOPUS:84867262719

SN - 1751-8113

VL - 45

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 41

M1 - 415101

ER -