Tarun K. Roy

Stability of invariant curves in four-dimensional reversible mappings near 1 : 1 resonance

Recent work on reversible Neimark-Sacker bifurcation in 4D maps is summarized. Linear stability analysis of families of invariant curves appearing in this bifurcation is presented by (a) referring to the analogous stability problem in reversible Hopt bifurcation in vector fields and (b) perturbatively calculating a set of quantities, termed quasi-multipliers, for the invariant curves. In particular, the critical rotation numbers corresponding to transition from elliptic to hyperbolic invariant curves on the subthreshold side in the so-called inverted bifurcation are calculated. Results of numerical iterations corroborating the above analysis are presented. The question of exploring the structure of the phase space close to the invariant curves is briefly addressed in the conclusion.

Effects of the Nature of Index Profile on the Validity of the Eikonal Approximation

Abstract We investigate the effects of the nature of the index profile of the scatterer on the validity of the eikonal approximation. For simplicity we concentrate on one-dimensional models where exact solutions can be obtained. Three kinds of scatterer are studied: (i) homogeneous with sharp boundaries (ii) inhomogeneous (continuously varying refractive index) and (iii) a model for rough surfaces. The relevance of such studies to the realistic case in three dimensions is pointed out.

Resonant collisions in four-dimensional reversible maps: a description of scenarios

Abstract We define a resonant collision of order k (≥1) in a family of four-dimensional (4D) reversible maps. For any specified k, the bifurcation scenario is the collection of the different possible types of bifurcation of a symmetric fixed point that may be encountered through various choices of parameters describing the family of maps under consideration. We adopt a perturbative approach, coupled with numerical iterations around orbits obtained perturbatively, to explore phase space structures in the immediate vicinity of a resonant collision and thereby to obtain a description of the possible scenarios for different values of k. The phase space structures typically involve bifurcating periodic orbits, families of invariant curves, and tori, and present interesting possibilities, especially around the ‘secondary bifurcations’ of the periodic orbits (see below). Based on the results of the perturbative and numerical approach we conjecture that three distinct scenarios are involved for the cases k = 2, 3, 4, respectively, while there exists a fourth distinctive scenario common to all k > 4, and we present what we believe to be a reasonably exhaustive description of these scenarios. The case k = 1 involves bifurcations of fixed points rather than of periodic orbits, and has been investigated numerically in an earlier paper.

Fourth order resonant collisions of multipliers in reversible maps: period-4 orbits and invariant curves

Resonant collision of multipliers at ±i of a symmetric fixed point for a 2-parameter family of 4-dimensional reversible maps is considered. Bifurcation of period-4 orbits from the fixed point and their linear stability characteristics are briefly reviewed. In one of the three possible types of bifurcation (see text), a small angle secondary collision of the Floquet multipliers of the bifurcating periodic orbit takes place, leading to the bifurcation of invariant curves from the orbit. The invariant curves are calculated in a perturbation scheme in the leading order of perturbation. The secondary bifurcation is found to be of superthreshold type. An interesting pattern in the vicinity of the resonant collision, involving families of invariant curves and 2-tori, emerges. Results of numerical iterations, corroborating the picture conjectured on the basis of perturbation calculations, are presented. Corresponding results on resonant collisions at −1 are briefly stated.

Hopf bifurcation in four-dimensional reversible maps and renormalisation equations

Exact scaling ratios are obtained for the renormalisation equations involving the doubling operator in reversible Hopf bifurcation in four-dimensional maps, and a scaling law for the average duration of the orbits near an unstable fixed point is determined. Numerical corroboration is presented.