Stochastic and Fractional Differential Equations: Attractor, Invariant Measure and Their Relationship

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 20 June 2024 | Viewed by 2391

Special Issue Editors


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Guest Editor
Department of Mathematics, Southwest Jiaotong University, Chengdu, China
Interests: random dynamical system; stochastic differential equations; attractor; invariant measure

E-Mail Website
Guest Editor
Department of Mathematics, Southwest Jiaotong University, Chengdu, China
Interests: stochastic differential equations; complex networks; stability; synchronization
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Special Issue Information

Dear Colleagues,

Mathematical models that study the evolution of many natural phenomena such as astroscience, fluid mechanics, plasma physics, and weather change are often nonlinear evolution equations and resulting infinite dimensional dynamic systems. However, in real life, the development of something is sometimes influenced by accidental random factors. Many studies have shown that due to the interaction between noise and nonlinearity, the system structure may be completely destroyed, making the system change from ordered to disordered, or vice versa. Therefore, it is necessary to study infinite dimensional random dynamic systems. The study of infinite dimensional random dynamical systems requires the combination of knowledge of dynamical systems, partial differential equations, fractional differential equations, functional analysis, stochastic analysis, and the complexity of their own problems. Currently, this is still in the initial and innovative stage.

The focus of this Special Issue is to continue to advance research on topics relating to the theory and application of infinite dimensional random dynamical systems. Topics that are invited for submission include (but are not limited to):

  1. Random attractors of stochastic and fractional differential equations;
  2. Invariant measures of stochastic and fractional differential equations;
  3. The relationship between random attractors and invariant measures;
  4. Stability of stochastic complex systems.

Prof. Dr. Dingshi Li
Dr. Chunmei Zhang
Guest Editors

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Keywords

  • random dynamical system
  • stochastic differential equation
  • fractional differential equation
  • random attractor
  • invariant measures
  • random fractional

Published Papers (2 papers)

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Research

18 pages, 351 KiB  
Article
The Effects of Nonlinear Noise on the Fractional Schrödinger Equation
by Jin Xie, Han Yang, Dingshi Li and Sen Ming
Fractal Fract. 2024, 8(1), 19; https://doi.org/10.3390/fractalfract8010019 - 26 Dec 2023
Viewed by 959
Abstract
The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. [...] Read more.
The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term. Full article
17 pages, 48667 KiB  
Article
Equilibrium Problem for the Stochastic Multi-Weighted Urban Public Transportation System with Time Delay: A Graph-Theoretic Method
by Hui Yang, Chunmei Zhang, Ran Li and Huiling Chen
Fractal Fract. 2023, 7(10), 767; https://doi.org/10.3390/fractalfract7100767 - 19 Oct 2023
Viewed by 1013
Abstract
This paper focuses on the equilibrium problem of an urban public transportation system with time delay. Time delay, multi-weights, and stochastic disturbances are considered in the urban public transportation system. Hence, one can regard the urban public transportation system as a stochastic multi-weighted [...] Read more.
This paper focuses on the equilibrium problem of an urban public transportation system with time delay. Time delay, multi-weights, and stochastic disturbances are considered in the urban public transportation system. Hence, one can regard the urban public transportation system as a stochastic multi-weighted delayed complex network. By combining graph theory and the Lyapunov method, the global Lyapunov function is constructed indirectly. Moreover, the response system can realize synchronization with the drive system under the adaptive controller. In other words, the urban public transportation system is balanced in the actual running traffic network. Finally, numerical examples about the Chua system and small-world network are presented to confirm the accuracy and validity of the theoretical results. Full article
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