Periodic and chaotic solutions for a nonlinear system arising from a nuclear spin generator

Abstract

We study in great detail a system of three first-order ordinary differential equations describing the behaviour of a nuclear spin generator (NSG). This system, not much referred to in literature, displays a large variety of behaviours, both regular and chaotic. Existence of periodic solutions is proved for certain ranges of parameters. Stability criteria for periodic solutions are given. The system is shown here to have a codimension-two bifurcation point: a Takens-Bogdanov bifurcation point. Chaotic behaviours arise from (i) period-doubling bifurcations, (ii) intermittency route, and (iii) homoclinic bifurcations. The gluing of strange attractors and their ungluing, with periodic and chaotic behaviours in the intervening parametric range, not reported earlier for any chaotic system, are shown to occur for the NSG system. Also, in certain parameteric intervals, coexisting attractors and coexisting strange attractors are found to occur. In view of the larger variety of phenomena exhibited by NSG in comparison with the Lorenz system, it is claimed that the former is a better archetypal system for chaos.