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Mathematics
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Infinite Dimensional Dynamical System and Differential Equations
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Infinite Dimensional Dynamical System and Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 March 2024) | Viewed by 8426

Special Issue Editor

Prof. Dr. Caidi Zhao

E-Mail Website
Guest Editor
Department of Mathematics, Wenzhou University, Wenzhou 325035, China
Interests: nonlinear partial differential equations; infinite dimensional dynamical system; lattice dynamical system; fluid dynamics

Special Issue Information

Dear Colleagues,

The study of dynamical systems and their application has a long history featuring many achievements and challenges, and continues to be among the mainstays of contemporary mathematics. Infinite-dimensional dynamical systems are generated by evolutionary equations describing the evaluations of systems whose states are in infinite-dimensional phase spaces. The aim of this Special Issue of Mathematics is to study the long-term behavior of several kinds of infinite-dimensional dynamical systems associated with differential equations. This Special Issue will be devoted to the subjects of differential equations, dynamical systems, infinite dimensional analyses, and their advanced applications.

Potential topics include but are not limited to:

  • Infinite dimensional dynamical systems;
  • Random dynamical systems;
  • Lattice dynamical systems;
  • Ordinary differential equations;
  • Delay differential equations;
  • Functional equations;
  • Partial differential equations;
  • Fractional differential equations;
  • Stochastic differential equations;
  • Integral equations;
  • Applications of fixed-point theorems to nonlinear equations.

Prof. Dr. Caidi Zhao
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
  • infinite-dimensions
  • ordinary differential equations
  • delay differential equations
  • functional equations
  • partial differential equations
  • fractional differential equations
  • stochastic differential equations
  • integral equations

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Published Papers (5 papers)

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Research

14 pages, 274 KiB  
Article
Large Time Decay Rates of the 2D Micropolar Equations with Linear Velocity Damping
by Jingbo Wu, Qing-Qing Wang and Tian-Fang Zou
Mathematics 2023, 11(10), 2311; https://doi.org/10.3390/math11102311 - 16 May 2023
Viewed by 903
Abstract
This paper studies the large time behavior of solutions to the 2D micropolar equations with linear damping velocity. It is proven that the global solutions have rapid time decay rates [...] Read more.
This paper studies the large time behavior of solutions to the 2D micropolar equations with linear damping velocity. It is proven that the global solutions have rapid time decay rates ∥∇ω∥L2+∥∇u∥L2≤C(1+t)−32 and ∥u∥L2≤C(1+t)−32,∥ω∥L2≤C(1+t)−1. The findings are mainly based on the new observation that linear damping actually improves the low-frequency effect of the system. The methods here are also available for complex fluid models with linear damping structures. Full article
(This article belongs to the Special Issue Infinite Dimensional Dynamical System and Differential Equations)
20 pages, 1511 KiB  
Article
Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation
by Meng Zhu, Jing Li and Xinze Lian
Mathematics 2022, 10(17), 3171; https://doi.org/10.3390/math10173171 - 3 Sep 2022
Cited by 4 | Viewed by 1749
Abstract
In this paper, we consider a Leslie–Gower cross diffusion predator–prey model with a strong Allee effect and hunting cooperation. We mainly investigate the effects of self diffusion and cross diffusion on the stability of the homogeneous state point and processes of pattern formation. [...] Read more.
In this paper, we consider a Leslie–Gower cross diffusion predator–prey model with a strong Allee effect and hunting cooperation. We mainly investigate the effects of self diffusion and cross diffusion on the stability of the homogeneous state point and processes of pattern formation. Using eigenvalue theory and Routh–Hurwitz criterion, we analyze the local stability of positive equilibrium solutions. We give the conditions of Turing instability caused by self diffusion and cross diffusion in detail. In order to discuss the influence of self diffusion and cross diffusion, we choose self diffusion coefficient and cross diffusion coefficient as the main control parameters. Through a series of numerical simulations, rich Turing structures in the parameter space were obtained, including hole pattern, strip pattern and dot pattern. Furthermore, We illustrate the spatial pattern through numerical simulation. The results show that the dynamics of the model exhibits that the self diffusion and cross diffusion control not only form the growth of dots, stripes, and holes, but also self replicating spiral pattern growth. These results indicate that self diffusion and cross diffusion have important effects on the formation of spatial patterns. Full article
(This article belongs to the Special Issue Infinite Dimensional Dynamical System and Differential Equations)
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15 pages, 350 KiB  
Article
Homogenization of Trajectory Statistical Solutions for the 3D Incompressible Micropolar Fluids with Rapidly Oscillating Terms
by Hujun Yang, Xiaoling Han and Caidi Zhao
Mathematics 2022, 10(14), 2469; https://doi.org/10.3390/math10142469 - 15 Jul 2022
Cited by 3 | Viewed by 1166
Abstract
This article studies the 3D incompressible micropolar fluids with rapidly oscillating terms. The authors prove that the trajectory statistical solutions of the oscillating fluids converge to that of the homogenized fluids provided that the oscillating external force and angular momentum possess some weak [...] Read more.
This article studies the 3D incompressible micropolar fluids with rapidly oscillating terms. The authors prove that the trajectory statistical solutions of the oscillating fluids converge to that of the homogenized fluids provided that the oscillating external force and angular momentum possess some weak homogenization. The results obtained indicate that the trajectory statistical information of the 3D incompressible micropolar fluids has a certain homogenization effect with respect to spatial variables. In addition, our approach is also valid for a broad class of evolutionary equations displaying the property of global existence of weak solutions without a known result of global uniqueness, including some model hydrodynamic equations, MHD equations and non-Newtonian fluids equations. Full article
(This article belongs to the Special Issue Infinite Dimensional Dynamical System and Differential Equations)
15 pages, 948 KiB  
Article
Behavior Analysis of a Class of Discrete-Time Dynamical System with Capture Rate
by Xiongxiong Du, Xiaoling Han and Ceyu Lei
Mathematics 2022, 10(14), 2410; https://doi.org/10.3390/math10142410 - 10 Jul 2022
Cited by 4 | Viewed by 1388
Abstract
In this paper, we study the stability and bifurcation analysis of a class of discrete-time dynamical system with capture rate. The local stability of the system at equilibrium points are discussed. By using the center manifold theorem and bifurcation theory, the conditions for [...] Read more.
In this paper, we study the stability and bifurcation analysis of a class of discrete-time dynamical system with capture rate. The local stability of the system at equilibrium points are discussed. By using the center manifold theorem and bifurcation theory, the conditions for the existence of flip bifurcation and Hopf bifurcation in the interior of R+2 are proved. The numerical simulations show that the capture rate not only affects the size of the equilibrium points, but also changes the bifurcation phenomenon. It was found that the discrete system not only has flip bifurcation and Hopf bifurcation, but also has chaotic orbital sets. The complexity of dynamic behavior is verified by numerical analysis of bifurcation, phase and maximum Lyapunov exponent diagram. Full article
(This article belongs to the Special Issue Infinite Dimensional Dynamical System and Differential Equations)
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7 pages, 766 KiB  
Article
On the Residual Continuity of Global Attractors
by Xingxing Wang and Hongyong Cui
Mathematics 2022, 10(9), 1444; https://doi.org/10.3390/math10091444 - 25 Apr 2022
Cited by 4 | Viewed by 1476
Abstract
In this brief paper, we studied the residual continuity of global attractors Aλ in varying parameters λ∈Λ with Λ a bounded Borel set in Rd. We first reviewed the well-known residual continuity result of global attractors and then [...] Read more.
In this brief paper, we studied the residual continuity of global attractors Aλ in varying parameters λ∈Λ with Λ a bounded Borel set in Rd. We first reviewed the well-known residual continuity result of global attractors and then showed that this residual continuity is equivalent to the dense continuity. Then, we proved an analogue continuity result in measure sense that, under certain conditions, the set-valued map λ↦Aλ is almost (in the Lebesgue measure sense) uniformly continuous: for any small ε>0 there exists a closed subset Cε⊂Λ with Lebesgue measure m(Cε)>μ(Λ)−ε such that the set-valued map ε↦Aε is uniformly continuous on Cε. This, in return, indicates that the selected attractors {Aλ:λ∈Cε} can be equi-attracting. Full article
(This article belongs to the Special Issue Infinite Dimensional Dynamical System and Differential Equations)
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