Some New Analytical and Numerical Results for the Nuclear Spin Generator (Sherman) System

This work studies the behavior of the nuclear spin generator (NSG) system, also known as the Sherman system. The analysis is carried out analytically and numerically by finding both exact solutions and bifurcations. We have obtained new analytical and numerical results that clarify the complex dynamics of the Sherman system depending on the values of involved parameters. We have established the possible equilibrium states and their nature, as well as conditions for the occurrence of bifurcations (local and global) and chaotic behavior upon reaching certain values of parameters. Three families of exact solutions to the studied autonomous dynamical system were obtained, which determine trajectories in the phase space that approach, as a limit, one of the system’s equilibrium points as time tends to infinity. Two pairs of first integrals of the Sherman system are present in two cases when certain parameters are equal to zero. In the general case, we provide an independent elementary proof that the Sherman system has no polynomial first integrals, complementing the stronger results reported in the literature on the subject.