Chaotic Response of a Self-Interacting Pseudo-Spin Model

A new model for a conservative semi-quantum system is presented to study quantum aspects of chaotic behavior. The model is made up of a pseudo-spin which interacts with an external field and the reaction field of the polarization of the pseudo-spin. The reaction field is super-imposed upon the external field in the equations of motion and it brings nonlinear terms. Two isolating integrals, whose intersection curve gives an orbit, are obtained analytically in a static external field case, and phase plane plots are drawn using the integrals. The phase plane plots have three elliptic fixed points and one hyperbolic fixed point for a strong reaction field case and it has two elliptic ones and no hyperbolic one for a weak reaction field case. Only one single finite separatrix emanates from the hyperbolic fixed point in the strong reaction field case. Chaotic motion is observed in a periodic external field case, and Poincaré mappings, Lyapunov exponents and power spectra of the polarization are calculated with the aid of computer simulations. The spectrum of 1-dimensional Lyapunov exponents takes a hyperbolic type ( +, 0, - ), namely the model has C-system like property.