J. K. Bhattacharjee

Renormalization group and Lienard systems of differential equations

Autonomous Lienard systems, which constitute a huge family of periodic motions, exhibit limit cycle behaviour in certain cases and centres in others. In the literature, the signature for the existence of these two different facets of periodic behaviour has been studied from different geometrical perspectives and not from a general viewpoint. Starting out from general considerations, we show in this work that a certain renormalization scheme is capable of unifying these two different aspects of periodic motion. We show that the renormalization group allows a unified analysis of the limit cycle and centre in a Lienard system of differential equations. While the approach is perturbative, it is possible to make a stronger statement in this regard. Two different classes of Lienard systems have been considered. The analysis provides clear insight into how the frequency gets corrected at different orders of perturbation as one flips the parity of the damping term.

Center or limit cycle: renormalization group as a probe

AbstractnBased on our studies done on two-dimensional autonomous systems, forced non-autonomousnsystems and time-delayed systems, we propose a unified methodology – that usesnrenormalization group theory – for finding out existence of periodic solutions in anplethora of nonlinear dynamical systems appearing across disciplines. The technique willnbe shown to have a non-trivial ability of classifying the solutions into limit cycles andnperiodic orbits surrounding a center. Moreover, the methodology has a definite advantagenover linear stability analysis in analyzing centers.n

Quasi‐viscous accretion flow – I. Equilibrium conditions and asymptotic behaviour

In a novel approach to studying viscous accretion flows, viscosity has been introduced as a perturbative effect, involving a first-order correction in the α-viscosity parameter. This method reduces the problem of solving a second-order non-linear differential equation (Navier–Stokes equation) to that of an effective first-order equation. Viscosity breaks down the invariance of the equilibrium conditions for stationary inflow and outflow solutions, and distinguishes accretion from wind. Under a dynamical systems classification, the only feasible critical points of this ‘quasi-viscous’ flow are saddle points and spirals. On large spatial scales of the disc, where a linearized and radially propagating time-dependent perturbation is known to cause a secular instability, the velocity evolution equation of the quasi-viscous flow has been transformed to bear a formal closeness with Schrodingers equation with a repulsive potential. Compatible with the transport of angular momentum to the outer regions of the disc, a viscosity-limited length-scale has been defined for the full spatial extent over which the accretion process would be viable.

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.

Renormalization Group as a Probe for Dynamical Systems

The use of renormalization group (RG) in the analysis of nonlinear dynamical problems has been pioneered by Goldenfeld and co-workers [1]. We show that perturbative renormalization group theory of Chen et al can be used as an effective tool for asymptotic analysis for various nonlinear dynamical oscillators. Based on our studies [2] done on two-dimensional autonomous systems, as well as forced non-autonomous systems, we propose a unified methodology – that uses renormalization group theory – for finding out existence of periodic solutions in a plethora of nonlinear dynamical systems appearing across disciplines. The technique will be shown to have a non-trivial ability of classifying the solutions into limit cycles and periodic orbits surrounding a center. Moreover, the methodology has a definite advantage over linear stability analysis in analyzing centers.

Renormalisation group and isochronous oscillations

AbstractWe show how the condition of isochronicity can be studied for two-dimensional systems innthe renormalisation group (RG) context. We find a necessary condition for thenisochronicity of the Cherkas system and another class of cubic system. While the necessaryncondition involves a relation among the coefficients of the cubic terms, for certainnchoices of parameters, we can explicitly predict the specific values of the parameters fornisochrony. Our conditions are satisfied by all the cases studied recently bynGhosexa0Choudhury and Guhaxa0[J. Phys. A: Math. Theor. 43, 125202 (2010)] andnBardet etxa0al.xa0[Bull. Sci. Math. 135, 230 (2011)]. Finally we use our RGnapproach to identify a family of nonlinear isochronous oscillators not reported earlier innthe literature.

STATPHYS-Kolkata VII

In the past two decades, a series of international conferences on Statistical Physics, going by the name Statphys Kolkata, have been organized in Kolkata (previously Calcutta) at roughly three-year intervals, the first one being held in 1992–93. The seventh of this series, Statphys Kolkata VII (http://www.saha.ac.in/cmp/stat.vii/index.php) was held from 26–30 November 2010. This meeting was organized as part of the Silver Jubilee Celebration of the Satyendra Nath Bose National Centre for Basic Sciences, Kolkata, in collaboration with the Saha Institute of Nuclear Physics, Kolkata. In Statphys Kolkata VII, a few topics of current interest such as Collective behavior and emergent phenomena, Systems far from equilibrium, Soft matter, and Quantum critical phenomena were given special emphasis, while various other issues of Statistical Physics were also addressed. We were happy to note that the conference attracted a large number of participants, and the talk and poster sessions generated a lot of discussions, arguments and collaborations. The articles appearing in this proceedings are based on the invited talks and selected poster presentations. We would like to thank the Journal of Physics Conference Series, IOP, for publishing the proceedings of the conference, and the referees for their prompt and active support. The proceedings of the earlier Statphys Kolkata conferences have appeared in Physica A, vol 384 (2007); Physica A, vol 346 (2005); Physica A, vol 318 (2003); Physica A, vol 270 (1999); Physica A, vol 224 (1996); and Physica A, vol 186 (1992). We would like to take this opportunity to thank all the members of the organizing committee (especially Dr Anjan Kumar Chandra for extensive all-round help), and acknowledge the Centre for Applied Mathematics and Computational Science (CAMCS, Saha Institute of Nuclear Physics, Kolkata) and Satyendra Nath Bose National Centre for Basic Sciences, Kolkata, for their financial support. Jayanta Kumar Bhattacharjee, Bikas K Chakrabarti, Jun-Ichi Inoue and Parongama SenConvenors and Editors of the proceedings