Fixed points of an operator associated with the Ising model on a closed Cayley tree
Annotatsiya
Abstract In this paper, we investigate the fixed points of a nonlinear operator arising from the Ising model on a closed Cayley tree with branching ratio k = 3. By analyzing the associated system of recursive equations, we derive explicit conditions on the interaction parameters under which the operator admits one, three, five, or at least seven fixed points. The analysis is carried out without imposing symmetry constraints on the interaction parameters, allowing for a comprehensive study of the asymmetric case. Using invariant submanifolds of the operator, we reduce the problem to lower-dimensional equations and obtain exact algebraic criteria for the multiplicity of solutions. Furthermore, we compute the corresponding local magnetizations associated with each fixed point and classify the resulting thermodynamic behaviors. In particular, we identify parameter regions corresponding to disordered phases with vanishing magnetization and ordered phases with nonzero magnetization, indicating phase coexistence. To complement the analytical results, we perform a stability analysis based on the eigenvalues of the Jacobian matrix and provide a numerical verification via the maximal Lyapunov exponent. The combined analytical and numerical results demonstrate that asymmetric interactions significantly enrich the phase structure of the model and lead to a complex organization of stable and unstable regimes.