<abstract xmlns="http://www.w3.org/1999/xhtml">

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 <italic>Mn(Fq)</italic> consisting of full matrices of order <italic>n</italic> over a finite field <italic>Fq</italic>. The second is the self-isomorphism of ideal relational graphs on a finite field <italic>Fq</italic>. It is proved that any self-isomorphism problem of graph <italic>Mn</italic>(<italic>Fq</italic>) when <italic>n</italic> >3 can be used with both criteria on it. Finally, a classical model of a dynamical system obtained from <italic>f</italic>(<italic>x</italic>) = cos <italic>x</italic> iterations is studied and its global convergence is discussed. Numerical experimental results show that the discrete dynamical system generated by function <italic>f</italic>(<italic>x</italic>) = cos <italic>x</italic> iteration has a unique <italic>ω</italic> 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.
</abstract>

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 Mn(Fq) consisting of full matrices of order n over a finite field Fq. The second is the self-isomorphism of ideal relational graphs on a finite field Fq. It is proved that any self-isomorphism problem of graph Mn(Fq) 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.

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

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

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...

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

Publicado en línea: 05 ago 2024

Páginas: -

Recibido: 23 abr 2024

Aceptado: 07 jul 2024

DOI: https://doi.org/10.2478/amns-2024-2105

Palabras clave
Classical dynamical systems, Algebraic structures, Ordered states, Self-isomorphism, Global convergence

© 2024 Lijuan Zhang, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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

Publicado en línea: 05 ago 2024

Páginas: -

Recibido: 23 abr 2024

Aceptado: 07 jul 2024

DOI: https://doi.org/10.2478/amns-2024-2105

Palabras claveClassical dynamical systems, Algebraic structures, Ordered states, Self-isomorphism, Global convergence

© 2024 Lijuan Zhang, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Palabras clave
Classical dynamical systems, Algebraic structures, Ordered states, Self-isomorphism, Global convergence