Differential Dynamical Systems Research Articles

Spectral bounds on the entropy flow rate and Lyapunov exponents in differentiable dynamical systems

Abstract Some microscopic dynamics are also macroscopically irreversible, dissipating energy and producing entropy. For many-particle systems interacting with deterministic thermostats, the rate of thermodynamic entropy dissipated to the environment is the average rate at which phase space contracts. Here, we use this identity and the properties of a classical density matrix to derive upper and lower bounds on the entropy flow rate with the spectral properties of the local stability matrix. These bounds are an extension of more fundamental bounds on the Lyapunov exponents and phase space contraction rate of continuous-time dynamical systems. They are maximal and minimal rates of entropy production, heat transfer, and transport coefficients set by the underlying dynamics of the system and deterministic thermostat. Because these limits on the macroscopic dissipation derive from the density matrix and the local stability matrix, they are numerically computable from the molecular dynamics. As an illustration, we show that these bounds are on the electrical conductivity for a system of charged particles subject to an electric field.

Solutions for a New Fractional Differential Dynamical System and Yosida Quasi-Inverse Variational Inequality in Hilbert Space

In this article, first we introduce and study a Yosida Quasi-inverse variational inequality problem (in short, YQIVI) in Hilbert space and then developed a new fractional differential dynamical system for the YQIVI. We prove the existence and uniqueness of solution for the suggested dynamical system. Further, using the Lyapunov function we also prove the asymptotic stability of the new dynamical system at the equilibrium point. Furthermore, using Rothe's time discretization method we investigate existence and uniqueness of solution of the proposed dynamical system. Finally, we provide a numerical example to demonstrate the credibility and efficacy of the dynamical system in solving the YQIVI.

Dual CAR-T cell therapy for glioblastoma: strategies to cure tumour diseases based on a mathematical model

AbstractThe CAR-T cell immunotherapy entails the genetic reprogramming of T-lymphocytes, which then engage with cancer cells, triggering an anti-tumour immune response. While this treatment has gained approval for hematological cancers, addressing solid tumours presents new obstacles. Challenges include the heterogeneity of antigen expression within solid tumours, encompassing antigen-positive non-tumoural cells, the presence of immune inhibitory molecules, and the difficulty of CAR-T cell trafficking within the tumour microenvironment. In this article, we analytically study a generalisation of a mathematical model proposed by León-Triana et al. (Cancers 13(4):703, 2021a. https://doi.org/10.3390/cancers13040703, Commun Nonlinear Sci Numer Simul 94:105570). This model focuses on the dynamics of glioblastoma, the most aggressive brain tumour, and its response to CAR-T cell treatment. We study the basic properties of the model, the dynamics of the solutions of the model when the treatment is not sustained during the time, and finally we study analytically the model when the therapy is constant, periodic and/or impulsive. We derive sufficient conditions for global stability of tumour-free equilibrium, as well as necessary and sufficient conditions for local stability of the equilibrium obtaining conditions for an effective treatment. Finally, we perform different numerical simulations to find the strategies to keep the tumour under control. The obtained results are based on a combination of different analytical techniques in differential equations, dynamical systems and numerical simulations.

Fixed Point Results for New Classes of k-Strictly Asymptotically Demicontractive and Hemicontractive Type Multivalued Mappings in Symmetric Spaces

Fixed point theory is a significant area of mathematical analysis with applications across various fields such as differential equations, optimization, and dynamical systems. Recently, multivalued mappings have gained attention due to their ability to model more complex and realistic problems. ln this work, novel classes of nonlinear mappings called k-strictly asymptotically demicontractive-type and asymptotically hemicontractive-type multivalued mappings are introduced in real Hilbert spaces that are symmetric spaces. In addition, we discuss the weak and strong convergence results by considered modified algorithms, and a demiclosedness property, for these classes of mappings are proved. Several non-trivial examples are demonstrated to validate the newly defined mappings. Consequently, the results and iterative methods obtained in this study improve and extend several known outcomes in the literature.

The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies

Gronwall's inequalities are important in the study of differential equations and integral inequalities. Gronwall inequalities are a valuable mathematical technique with several applications. They are especially useful in differential equation analysis, stability research, and dynamic systems modeling in domains spanning from science and math to biology and economics. In this paper, we present new generalizations of Gronwall inequalities of integral versions. The proposed results involve $( \rho ,\varphi)-$Riemann-Liouville fractional integral with respect to another function. Some applications on differential equations involving $( \rho ,\varphi)-$Riemann-Liouville fractional integrals and derivatives are established.

The fluctuational transition mechanism of non-hyperbolic chaotic invariant sets

In order to reveal the general escape mechanism of non-hyperbolic chaotic invariant sets under both weak noise limit and finite noise intensity, the simplest example of Hénon map, which represents the stretching and folding of the phase space, is taken to study the escape mechanism under two kinds of global bifurcation: the fractal boundary crisis and the attractor contact crisis. In this paper, we revealed the general exit mechanism by analyzing the escape paths derived from the shooting method and Monte Carlo simulation. Finally, to further demonstrate universality, an example of high-dimensional differential dynamical system, namely the transient chaos Duffing oscillator, is examined, which underscores the main idea of this paper analyzing specific deterministic structures not only enhances the comprehensibility of the escape process, but also allows for predictions of general escape behavior under weak noise intensity.

Data-Driven Inverse Reinforcement Learning Control for Linear Multiplayer Games.

This article proposes a data-driven inverse reinforcement learning (RL) control algorithm for nonzero-sum multiplayer games in linear continuous-time differential dynamical systems. The inverse RL problem in the games is solved by a learner reconstructing the unknown expert players' cost functions from demonstrated expert's optimal state and control input trajectories. The learner, thus, obtains the same control feedback gains and trajectories as the expert, only using data along system trajectories without knowing system dynamics. This article first proposes a model-based inverse RL policy iteration framework that has: 1) policy evaluation step for reconstructing cost matrices using Lyapunov functions; 2) state-reward weight improvement step using inverse optimal control (IOC); and 3) policy improvement step using optimal control. Based on the model-based policy iteration algorithm, this article further develops an online data-driven off-policy inverse RL algorithm without knowing any knowledge of system dynamics or expert control gains. Rigorous convergence and stability analysis of the algorithms are provided. It shows that the off-policy inverse RL algorithm guarantees unbiased solutions while probing noises are added to satisfy the persistence of excitation (PE) condition. Finally, two different simulation examples validate the effectiveness of the proposed algorithms.

Global Exponential Robust Stability of Generalized Differential Dynamical System With Deviating Argument and Random Disturbance

Global Exponential Robust Stability of Generalized Differential Dynamical System With Deviating Argument and Random Disturbance

Fourier neural operator for real-time simulation of 3D dynamic urban microclimate

Global urbanization has underscored the significance of urban microclimates for human comfort, health, and building/urban energy efficiency. However, analyzing urban microclimates requires considering a complex array of outdoor parameters within computational domains at the city scale over a longer period than indoors. As a result, numerical methods like Computational Fluid Dynamics (CFD) become computationally expensive when evaluating the impact of urban microclimates. The rise of deep learning techniques has opened new opportunities for accelerating the modeling of complex nonlinear interactions and system dynamics. Recently, the Fourier Neural Operator (FNO) has been shown to be very promising in accelerating solving the Partial Differential Equations (PDEs) and modeling fluid dynamic systems. In this work, we apply the FNO network for real-time three-dimensional (3D) urban microclimate simulation. For modeling large-scale urban microclimate problems, CityFFD simulates urban microclimate features based on the semi-Lagrangian approach and fractional stepping method with the Smagorinsky large eddy simulation model. In our simulation, the 1200 sequential time steps are used as training data. We retain and analyze the data from all stages, including the spin-up period, because we wish to understand how the flow develops transiently from initial conditions, and both one-step and sequential timestep predictions are analyzed. When applied to unseen data with different wind directions, the FNO model has a 0.3% one-step prediction error and a maximum error of 5%. A real-time simulation of urban microclimates in 3D is possible with the FNO approach, which is 25 times faster than the traditional numerical solver.

Trajectory controllability of neutral stochastic integrodifferential equations with mixed fractional Brownian motion

This work focuses on the existence and trajectory (T-)controllability of mixed fractional Brownian motion (fBm) with the Hurst index and neutral stochastic integrodifferential equations (NSIDEs) with deviating argument and fBm. Stochastic integrodifferential equations (SIDEs) are solved in Hilbert space using stochastic analysis, the resolvent operator, and Krasnoselskii's fixed point theorem (KFPT). Furthermore, providing adequate assumptions, the T-controllability of the considered system is organised by using extended Gronwall's inequality. We demonstrate the theoretical insights and numerical simulations are included which is unique and makes this work more interesting. The obtained results generalise existing results from [Chalishajar, D. N., George, R. K., & Nandakumaran, A. K. (2010). Trajectory controllability of nonlinear integro-differential system. Journal of Franklin Institute, 347(7), 1065–1075.; Durga, N., Muthukumar, P., & Malik, M. (2022). Trajectory controllability of Hilfer fractional neutral stochastic differential equation with deviated argument and mixed fractional Brownian motion. Optimisation, 1–27.; Muslim, M., & George, R. K. (2019). Trajectory controllability of the nonlinear systems governed by fractional differential equations. Differential Equations and Dynamical Systems, 27, 529–537.; Dhayal, R., Malik, M., & Abbas, S. (2021). Approximate and trajectory controllability of fractional stochastic differential equation with non-instantaneous impulses and Poisson jumps. Asian Journal of Control, 23(6), 2669–2680.].

Noise-driven signal study of power systems based on stochastic partial differential equations

The exploration of stochastic partial differential equations in noisy perturbations of dynamical systems remains a major challenge at this stage. The study analyzes the effective dynamical system combining degenerate additive noise-driven stochastic partial differential equations, firstly in the first class of stochastic partial differential equations, the terms in the non-nuclear space formed by nonlinear interactions are overcome by effectively replacing the elements in the non-nuclear space through the ItÔ formulation, and thus the final effective dynamical system is obtained. The effective dynamical system is then obtained in the second type of stochastic partial differential equation using the O-U process similar to the terms in the non-nuclear space. At noise disturbance amplitudes of 5%, 10%, 15% and 20% AC voltage maxima in that order, the effective dynamical systems for the first type of stochastic partial differential equation and the second type of stochastic partial differential equation are more stable compared to the other types of partial differential equation dynamical systems, with the maximum range of error rate improvement for the sampling points 0–239 voltage rms and voltage initial phase value being 3.62% and 26.85% and 2.13% and 19.86% for sampling points 240–360, respectively. The effective dynamic system and stochastic partial differential equation obtained by the research have very high approximation effect, and can be applied to mechanical devices such as thermal power machines.

Classical Fisher information for differentiable dynamical systems.

Fisher information is a lower bound on the uncertainty in the statistical estimation of classical and quantum mechanical parameters. While some deterministic dynamical systems are not subject to random fluctuations, they do still have a form of uncertainty. Infinitesimal perturbations to the initial conditions can grow exponentially in time, a signature of deterministic chaos. As a measure of this uncertainty, we introduce another classical information, specifically for the deterministic dynamics of isolated, closed, or open classical systems not subject to noise. This classical measure of information is defined with Lyapunov vectors in tangent space, making it less akin to the classical Fisher information and more akin to the quantum Fisher information defined with wavevectors in Hilbert space. Our analysis of the local state space structure and linear stability leads to upper and lower bounds on this information, giving it an interpretation as the net stretching action of the flow. Numerical calculations of this information for illustrative mechanical examples show that it depends directly on the phase space curvature and speed of the flow.

New results of uncertain integrals and applications

Abstract Based on the uncertainty theory, Liu [B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst. 3 2009, 1, 3–10] introduced an uncertain integral for applying uncertain differential equation, finance, control, filtering and dynamical systems. Since uncertain integrals are the important content of uncertainty theory, this paper explores an approach of the relationship between uncertain integrals by the well-known Chebyshev-type inequality. Also, we propose the concept of an uncertain fractional integral which is generalized version of an uncertain integral. The definition of a strong comonotonic uncertain process and some new properties of the uncertain integral were presented in [C. You and N. Xiang, Some properties of uncertain integral, Iran. J. Fuzzy Syst. 15 2018, 2, 133–142]. Based on the strong comonotonic uncertain process, as an application, we provide Chebyshev’s inequality for a fractional uncertain integral and an uncertain integral.

Social Bots and Information Propagation in Social Networks: Simulating Cooperative and Competitive Interaction Dynamics

With the acceleration of human society’s digitization and the application of innovative technologies to emerging media, popular social media platforms are inundated by fresh news and multimedia content from multiple more or less reliable sources. This abundance of circulating and accessible information and content has intensified the difficulty of separating good, real, and true information from bad, false, and fake information. As it has been proven, most unwanted content is created automatically using bots (automated accounts supported by artificial intelligence), and it is difficult for authorities and respective media platforms to combat the proliferation of such malicious, pervasive, and artificially intelligent entities. In this article, we propose using automated account (bots)-originating content to compete with and reduce the speed of propagating a harmful rumor on a given social media platform by modeling the underlying relationship between the circulating contents when they are related to the same topic and present relative interest for respective online communities using differential equations and dynamical systems. We studied the proposed model qualitatively and quantitatively and found that peaceful coexistence could be obtained under certain conditions, and improving the controlled social bot’s content attractiveness and visibility has a significant impact on the long-term behavior of the system depending on the control parameters.

A NOVEL 5D SYSTEM GENERATED INFINITELY MANY HYPERCHAOTIC ATTRACTORS WITH THREE POSITIVE LYAPUNOV EXPONENTS

Little seems to be known about the five-dimensional (5D) differential dynamical system with infinitely many hyperchaotic attractors, which have three positive Lyapunov exponents under no or infinitely many equilibria. This article presents a 5D dynamical system that can generate infinitely many hyperchaotic attractors. Of particular interest is the system exists not only infinitely many hyperchaotic attractors but also infinitely many periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many non-hyperbolic equilibria, (iii) only infinitely many hyperbolic equilibria. By numerical analysis, one finds the 5D system could generate infinitely many coexisting hyperchaotic or chaotic or periodic attractors in the three kinds of equilibria cases. And one obtains the global dynamical behavior of the system, such as the Lyapunov exponential spectrum, bifurcation diagram. To study the hyperchaotic complexity of the 5D system, we rigorously show the stability of hyperbolic equilibria and some mathematical characterization for 5D Hopf bifurcation. In particular, the existence of an infinite number of isolated bifurcated periodic orbits is strictly proven. These complex dynamics studies in this paper may further contribute to a deep understanding of the hyperchaotic systems with infinitely many attractors.

Unsupervised Legendre–Galerkin Neural Network for Solving Partial Differential Equations

In recent years, machine learning methods have been used to solve partial differential equations (PDEs) and dynamical systems, leading to the development of a new research field called scientific machine learning, which combines techniques such as deep neural networks and statistical learning with classical problems in applied mathematics. In this paper, we present a novel numerical algorithm that uses machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unsupervised machine learning</i> algorithm that learns <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multiple instances</i> of the solutions for different types of PDEs. Our approach addresses the limitations of both data-driven and physics-based methods. We apply the proposed neural network to general 1D and 2D PDEs with various boundary conditions, as well as convection-dominated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">singularly perturbed PDEs</i> that exhibit strong boundary layer behavior.

Mathematical Aspects of General Relativity

General relativity is an area that naturally combines differential geometry, partial differential equations, global analysis and dynamical systems with astrophysics, cosmology, high energy physics, and numerical analysis. It is rapidly expanding and has witnessed remarkable developments in recent years.

Efficiency and stability analysis on nonlinear differential dynamical systems

The principle goal of the paper is to present proficient limited contrast finite difference schemes to execute on the nonlinear coupled partial differential system which emulate the overseeing differential framework. In this paper, more consideration is given to the exactness and security of the proposed numerical schemes by review consistency and union of the arrangement which can be seen from figures and information tables. For the nonlinear differential system, mesh independent results are expensive which are accommodated by the generation of block tridiagonal matrix structures (inherent properties of schemes) which are measured in terms of [Formula: see text] &amp; [Formula: see text] norms which lead to a superb concurrence with the investigative arrangement.

Automated identification of dominant physical processes

The identification of processes that locally and approximately dominate dynamical system behavior has enabled significant advances in understanding and modeling nonlinear differential dynamical systems. Conventional methods of dominant process identification involve piecemeal and ad hoc (non-rigorous, informal) scaling analyses to identify dominant balances of governing equation terms and to delineate the spatiotemporal boundaries (boundaries in space and/or time) of each dominant balance. For the first time, we present an objective global measure of the fit of dominant balances to observations, which is desirable for automation, and was previously undefined. Furthermore, we propose a formal definition of the dominant balance identification problem in the form of an optimization problem. We show that the optimization can be performed by various machine learning algorithms, enabling the automatic identification of dominant balances. Our method is algorithm agnostic and it eliminates reliance upon expert knowledge to identify dominant balances which are not known beforehand.

Classification and Soliton for a Generalized Fourth-Order Dispersive Nonlinear Schrödinger Equation in a Heisenberg Spin Chain

The main objective of this paper is to study bifurcations and solitons for a generalized fourth-order dispersive nonlinear Schrödinger equation in a Heisenberg spin chain via the theory of differential dynamical systems. By introducing the complex traveling wave transformation, we obtain the singular traveling wave system and the Hamiltonian function. By applying the classical analysis method, we investigate the bifurcations thoroughly and obtain the phase portraits of the bifurcations in four cases. Based on the phase portraits, we obtain a large number of analytic parametric representations for bounded solitons, including periodic wave solutions, solitary wave solutions, kink (anti-kink) wave solutions, compacton solutions, and a special class of bounded soliton solutions, corresponding to stable and unstable manifolds of the saddle point. These analytic parametric representations enhance the understanding of dynamic properties and characteristics of soliton solutions.