Exploring the phase transition challenge by analyzing stability in a 5-D dynamical system linked to (2,1/2)-MSIM

In this short research, we delve into the phase transition phenomenon by analyzing the stability of the dynamical system associated with the (2,1/2)-mixed spin Ising model on a Cayley tree of order three. Our analysis focuses on examining the five-dimensional dynamical system linked to the Ising model featuring mixed spin-$(2,1/2)$ setups, operating within a third-order Cayley tree. By scrutinizing the Jacobian matrix of this nonlinear dynamic system, we pinpoint the repelling fixed points, corresponding to the Gibbs measures tied to the given model. The existence of these repelling fixed points enables us to predict potential phase transitions in the model by identifying any additional fixed points. We also identify the areas where the model exhibits chaotic tendencies through an examination of the Lyapunov exponent. Additionally, we seek to understand whether the order of the tree impacts the chaotic behavior observed in the dynamic system.