Fixed points and stability of an operator for the SOS model with competing interactions on the binary tree

We study the fixed points and stability of a nonlinear operator associated with the three-state SOS model with one-level competing interactions on the binary Cayley tree. We provide an exact characterization of the parameter regions corresponding to one through seven positive fixed points of the associated nonlinear operator. For the physically relevant case in which the competing interaction coincides with the nearest-neighbour interaction, we determine the critical values separating different phase transition regimes and obtain a complete classification of the positive fixed points. We further investigate their local stability by analyzing the eigenvalues of the Jacobian matrix and classify the fixed points as attracting, repelling, or saddle. The analytical results are complemented by phase portraits illustrating the qualitative dynamics of the nonlinear operator.

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