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Advances in Nonlinear Dynamical Systems of Mathematical Physics
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Advances in Nonlinear Dynamical Systems of Mathematical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: closed (20 May 2025) | Viewed by 625

Special Issue Editor

Dr. Ignacio S. Gomez

E-Mail Website
Guest Editor
Department of Exact and Natural Sciences, State University of Southwest Bahia, Itapetinga 45700-000, Brazil
Interests: dynamical systems; statistical mechanics; information geometry; chaos theory; quantum mechanics; mathematical physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear dynamical systems have been a fundamental topic of study over the last few centuries due to the multiple advances in science originating from their vast and rigorous formulations of celestial mechanics, chaos dynamics, fractal geometry, emergence of complexity, neuroscience, and biological models, among others. In parallel to this progress, modern mathematical tools have been developed that provide bridges between measure theory, topology, functional analysis, group theory, and differential equations.

This Special Issue invites authors to submit original works and new research results about nonlinear dynamical systems of mathematical physics. Submissions addressing mathematical features of nonlinear dynamical systems bringing novel connections with foundations of physical theories are especially welcome.

Special topics include (but are not limited to) the following:

  • Nonlinear dynamical systems;
  • Ergodic hierarchy;
  • Chaotic systems;
  • Probabilistic description of dynamical systems;
  • Fractal aspects of nonlinear dynamics;
  • Complex systems;
  • Numerical simulation of nonlinear dynamics.

Dr. Ignacio S. Gomez
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • dynamical systems
  • nonlinear dynamics
  • chaos theory
  • mathematical physics
  • fractality
  • complexity

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Published Papers (1 paper)

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Research

22 pages, 1130 KiB  
Article
Two-Mode Hereditary Model of Solar Dynamo
by Evgeny Kazakov, Gleb Vodinchar and Dmitrii Tverdyi
Mathematics 2025, 13(10), 1669; https://doi.org/10.3390/math13101669 - 20 May 2025
Viewed by 80
Abstract
The magnetic field of the Sun is formed by the mechanism of hydromagnetic dynamo. In this mechanism, the flow of the conducting medium (plasma) of the convective zone generates a magnetic field, and this field corrects the flow using the Lorentz force, creating [...] Read more.
The magnetic field of the Sun is formed by the mechanism of hydromagnetic dynamo. In this mechanism, the flow of the conducting medium (plasma) of the convective zone generates a magnetic field, and this field corrects the flow using the Lorentz force, creating feedback. An important role in dynamo is played by memory (hereditary), when a change in the current state of a physical system depends on its states in the past. Taking these effects into account may provide a more accurate description of the generation of the Sun’s magnetic field. This paper generalizes classical dynamo models by including hereditary feedback effects. The feedback parameters such as the presence or absence of delay, delay duration, and memory duration are additional degrees of freedom. This can provide more diverse dynamic modes compared to classical memoryless models. The proposed model is based on the kinematic dynamo problem, where the large-scale velocity field is predetermined. The field in the model is represented as a linear combination of two stationary predetermined modes with time-dependent amplitudes. For these amplitudes, equations are obtained based on the kinematic dynamo equations. The model includes two generators of a large-scale magnetic field. In the first, the field is generated due to large-scale flow of the medium. The second generator has a turbulent nature; in it, generation occurs due to the nonlinear interaction of small-scale pulsations of the magnetic field and velocity. Memory in the system under study is implemented in the form of feedback distributed over all past states of the system. The feedback is represented by an integral term of the type of convolution of a quadratic form of phase variables with a kernel of a fairly general form. The quadratic form models the influence of the Lorentz force. This integral term describes the turbulent generator quenching. Mathematically, this model is written with a system of integro-differential equations for amplitudes of modes. The model was applied to a real space object, namely, the solar dynamo. The model representation of the Sun’s velocity field was constructed based on helioseismological data. Free field decay modes were chosen as components of the magnetic field. The work considered cases when hereditary feedback with the system arose instantly or with a delay. The simulation results showed that the model under study reproduces dynamic modes characteristic of the solar dynamo, if there is a delay in the feedback. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamical Systems of Mathematical Physics)
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