A comparative study of the homotopy analysis method and an improved renormalization group method is presented in the context of the Rayleigh and the Van der Pol equations. Efficient approximate formulae as functions of the nonlinearity parameter ε for the amplitudes a(ε) of the limit cycles for both these oscillators are derived. The improvement in the renormalization group analysis is achieved by invoking the idea of nonlinear time that should have significance in a nonlinear system. Good approximate plots of limit cycles of the concerned oscillators are also presented within this framework.

Parent-to-parent counseling involves the interaction of two groups of parents, senior and junior, whose children suffer from cerebral palsy with multiple disabilities. The main theme of the interaction is the experience accumulated concerning the progress of the treatment of their child, the understanding of parental responsibility, the scope of rehabilitation, and the change of attitude towards their disabled children during the treatment, management and rehabilitation process, under the supervision of a psychologist or rehabilitation psychologist. The aim of the study is an attempt to minimize the gap between caregivers and the affected child through an exchange of knowledge and experience within the therapeutic group interactions. The parents who have been attending the outpatient department at the National Institute for the Orthopaedically Handicapped for a couple of months take a leading role in this interaction program and newcomers benefit through empathetically realizing the therapeutic aspects, and, simultaneously, releasing their internal tensions and anxiety regarding the rehabilitation of their disabled children. The study has been conducted on 50 Bengali speaking parents, who have attended the parent-to-parent interaction program. They were all from rural, urban and city areas within a 150-km radius of the place of study in Kolkata, West Bengal, situated in the eastern part of India. They were all from lower-income group (60%) and middle-income group (40%) families. Their mean educational qualification was XII standard. All the cases were selected from the mild to moderate range of child disability. All the participants indicated the effectiveness of these therapeutic interaction programs, with an average of 80% of parents favoring this program and acknowledging it as most effective.

Phase space method provides a novel way for deducing qualitative features of nonlinear differential equations without actually solving them. The method is applied here for analyzing stability of circular orbits of test particles in various physically interesting environments. The approach is shown to work in a revealing way in Schwarzschild spacetime. All relevant conclusions about circular orbits in the Schwarzschild-de Sitter spacetime are shown to be remarkably encoded in a single parameter. The analysis in the rotating Kerr black hole readily exposes information as to how stability depends on the ratio of source rotation to particle angular momentum. As a wider application, it is exemplified how the analysis reveals useful information when applied to motion in a refractive medium, for instance, that of optical black holes.