A new approach to solving isomorphism problems of classical dynamical systems using algebraic structures

There exists a fixed rule in classical dynamical systems that describes a point in geometric space over time. In this paper, based on the algebraic structure perspective, the dynamical system is defined as a category characterized by ordered state projections, and the dynamical system is inscribed using the algebraic structure, covering the phase space, continuous self-maps containing a single parametric variable, and the dynamical system itself. Meanwhile, two types of self-isomorphisms of algebraic maps are explored. One is the self-isomorphism of ideal inclusion maps on an algebra M_n(F_q) consisting of full matrices of order n over a finite field F_q. The second is the self-isomorphism of ideal relational graphs on a finite field F_q. It is proved that any self-isomorphism problem of graph M_n(F_q) when n >3 can be used with both criteria on it. Finally, a classical model of a dynamical system obtained from f(x) = cos x iterations is studied and its global convergence is discussed. Numerical experimental results show that the discrete dynamical system generated by function f(x) = cos x iteration has a unique ω limit point of 0.735, indicating that the stability and predictability of classical dynamical systems can be achieved using algebraic structures, as well as revealing the complexity, instability, and chaos of the system.