Stability Of Dynamical Systems Research Articles

Study on the Solutions of Impulsive Integrodifferential Equations of Mixed Type Based on Infectious Disease Dynamical Systems

Since ancient times, infectious diseases have been a major source of harm to human health. Therefore, scientists have established many mathematical models in the history of fighting infectious diseases to study the law of infection and then analyzed the practicability and effectiveness of various prevention and control measures, providing a scientific basis for human prevention and research of infectious diseases. However, due to the great differences in the transmission mechanisms and modes of many diseases, there are many kinds of infectious disease dynamic models, which make the research more and more difficult. With the continuous progress of infectious disease research technology, people have adopted more ways to prevent and interfere with the derivation and spread of infectious disease, which will make the state of infectious disease system change in an instant. The mutation of this state can be described more scientifically and reasonably by the mathematical impulse dynamic system, which makes the research more practical. Based on this, a time-delay differential system model of infectious disease under impulse effect was established by means of impulse differential equation theory. A class of periodic boundary value problems for impulsive integrodifferential equations of mixed type with integral boundary conditions was studied. The existence of periodic solutions of these equations was obtained by using the comparison theorem, upper and lower solution methods, and the monotone iteration technique. Finally, combined with the practical application, the established time-delay differential system model was applied to the prediction of the stability and persistence of the infectious disease dynamic system, and the correctness of the conclusion was further verified. This study provides some reference for the prevention and treatment of infectious diseases.

Enhancing disturbance rejection in offshore drilling platforms: a dynamic positioning control scheme using equivalent-input-disturbance method with separated observers

This paper presents a dynamic positioning control scheme for offshore drilling platforms using the equivalent-input-disturbance (EID) method with separated observers. The scheme incorporates observer-based disturbance compensation, wave-filtering state estimation, input transformation, and state feedback control to achieve robust control performance. The main contributions include the design of an EID scheme with separated observers, enabling enhanced disturbance-rejection performance, incorporating a notch filter in the EID estimator to reduce energy loss, and giving an observer tuning scheme to achieve a balanced performance between disturbance rejection and wave filtering. Simulation results demonstrate the effectiveness of the proposed scheme. This work provides valuable insights into designing stable and efficient dynamic positioning systems for offshore drilling platforms.

Generalization of KCC-theory to fractional dynamical systems and application to viscoelastic oscillations

As a geometrical method for analyzing the stability of dynamical systems, Kosambi–Cartan–Chern (KCC)-theory determines the Jacobi stability of a perturbation given to a trajectory based on the deviation curvature tensor. In this study, KCC-theory is generalized to dynamical systems with fractional derivatives, and the deviation curvature tensor is obtained from the fractional differential equation under the condition that the order of the fractional derivatives is close to an integer. Fractional KCC-theory is then applied to a one-dimensional spring–mass–damper model in which the damping force is expressed as a fractional derivative. In this case, three solution types are obtained depending on the damping coefficient and the fractional derivative order. One is damped oscillation with overshoot, which also appears in models with integer order derivatives, and another is damped oscillation without overshoot. The remaining one is overdamping, which neither oscillates nor overshoots. The deviation curvature tensor obtained from the fractional oscillatory model shows that both the damped oscillation without overshoot and the overdamping are Jacobi stable. When a discontinuous Jacobi instability appears on the solution curve, the solution type is classified as damped oscillation with overshoot. This result indicates that damped oscillation with overshoot and other solutions can be classified according to Jacobi stability.

Asymptotical Stabilization of Logic Dynamical Systems via Output-Based Random Control

Asymptotical Stabilization of Logic Dynamical Systems via Output-Based Random Control

Динамическая стабильность как категория диалектического взаимодействия хаоса и космоса в онтологии сетевого бытия

The increasing information flow has led to, inter alia, the dominant role of networking in communications. Devirtualization of networking in information and physical networks has made it possible to introduce network discourse into modern philosophy. The study of writings by Laozi, Confucius and their followers demonstrates that network discourse and network thinking were developed in the philosophical works of ancient Chinese scholars as representatives of Eastern philosophical discourse. At the same time, Western philosophical discourse, which can be traced to Plato and Aristotle, rests on linear logic and linear thinking. It is only towards the end of the 20th century that the linear/network thinking opposition weakened with the recognition of network interaction, under the influence of a profound transformation during the transition to a new stage of civilization’s development, that is, to the information/digital society. This can be illustrated by the evolution of the perception of the categories chaos and cosmos in Western (linear) and Eastern (network) philosophical discourse as the basis of the development of the modern representation of dynamic stability of the network interaction system. Using the information and communication method, this paper analyses Western and Eastern philosophical discourse from the Axial Age to the present day and substantiates the introduction of the category of dynamic stability. The latter stands for a dialectically connected process of creation/destruction of communications between actors of a network interaction under the influence of the dynamics of information exchange, which indicates that a network interaction system is autopoietic. Further development of the concept of dynamic stability requires creating a mathematical apparatus for calculating measures, that is, information entropy, of a dynamically stable network. These purely abstract studies are expected to be of practical importance for the development of social networks of interpersonal and intergroup communication.

An approach to nonlinear optimisation via set stabilisation of dynamical systems with an application to synchronous machines

An approach to constraint nonlinear optimisation is proposed, where the variables of the objective function and of the constraint function belong to the state of a nonlinear dynamical system. A nonlinear closed-loop feedback system is designed, whose solutions converge to a target set consisting of the optimal points of the underlying optimisation problem. The feedback law for the control input of the nonlinear dynamical system is designed by exact linearisation of a dynamical formulation of a first-order optimality condition. For the problem formulation and the stability analysis of the nonlinear closed-loop feedback system, the notion of V-detectability is considered. An application to energy-optimal feed-forward torque control for different types of synchronous machines is presented, which demonstrates the usefulness of the method. In this regard, anisotropic permanent magnet synchronous machines and electrically excited synchronous machines are covered.

Exploring electronic features in monolayer and bilayer MX2 (M = Hf, Zr; X = S, Se) structures under shear strain

This study employed first-principles calculations to investigate the dynamic stability of monolayer, bilayer, and heterostructure systems of four MX2 (M = Hf, Zr and X = S, Se) TMDCs, as well as the influence of shear strain on their electronic properties. The computed binding energies and phonon dispersions demonstrate the excellent dynamic stability of bilayer systems, while the small effective masses of electrons and holes in monolayer systems suggest high carrier mobility. Under shear strain, the bandgap size and type of monolayer MX2 system and their vdW heterostructures are effectively modulated, transitioning to metallic behavior under certain strain conditions. Both types of heterostructures exhibit type-II band alignment, serving as direct bandgap semiconductors, facilitating effective electron-hole separation between different layers and thereby reducing recombination. This study elucidates the tunability of the electronic properties of monolayers, bilayers, and heterostructures of MX2 systems under shear strain, which is important for expanding material functionalities and optimizing device performance.

Assessment of spilled oil dispersion affected by dispersant: Characteristic, stability, and related mechanism

Oil dispersion, a crucial process in oil transport, involves the detachment of oil droplets from slicks and their introduction into the water column, influencing subsequent oil migration and transformation. This study examines oil dispersion, considering characteristics, stability, and mechanisms, while evaluating the impact of dispersants and salinity. Results show the significant role of surfactant type in dispersants on oil dispersion characteristics, with anionic surfactants exhibiting higher sensitivity to salinity changes compared to nonionic surfactants. The dispersion efficiency varies with salinity, with anionic surfactants performing better in low salinity (<20‰) and nonionic surfactants showing superior performance at 30–35‰ salinities. Rheological analysis illustrates the breakup and coalescence of oil droplets within the shear rates of breaking waves. An increase in interfacial film rigidity impedes the coalescence of oil droplets, contributing to the dynamic stability of the oil-water hybrid system. The use of GM-2, a nonionic dispersant, results in the formation of a solid-like interface, characterized by increased elastic modulus, notably at 20‰ salinity. However, stable droplet size distribution (DSD) at 35‰ salinity for 60 h suggests droplets can remain dispersed in seawater. The enhancement of stability of oil dispersion is interpreted as the result of two mechanisms: stabilizing DSD and developing the strength of viscoelastic interfacial film. These findings offer insights into oil dispersion dynamics, highlighting the importance of surfactant selection and salinity in governing dispersion behavior, and elucidating mechanisms underlying dispersion stability.

Generalized quadratic embeddings for nonlinear dynamics using deep learning

The engineering design process often relies on mathematical modeling that can describe the underlying dynamic behavior. In this work, we present a data-driven methodology for modeling the dynamics of nonlinear systems. To simplify this task, we aim to identify a coordinate transformation that allows us to represent the dynamics of nonlinear systems using a common, simple model structure. The advantage of a common simple model is that customized design tools developed for it can be applied to study a large variety of nonlinear systems. The simplest common model – one can think of – is linear, but linear systems often fall short in accurately capturing the complex dynamics of nonlinear systems. In this work, we propose using quadratic systems as the common structure, inspired by the lifting principle. According to this principle, smooth nonlinear systems can be expressed as quadratic systems in suitable coordinates without approximation errors. However, finding these coordinates solely from data is challenging. Here, we leverage deep learning to identify such lifted coordinates using only data, enabling a quadratic dynamical system to describe the system’s dynamics. Additionally, we discuss the asymptotic stability of these quadratic dynamical systems. We illustrate the approach using data collected from various numerical examples, demonstrating its superior performance with the existing well-known techniques.

Distributed Time-Varying Optimization-An Output Regulation Approach.

This article deals with distributed algorithm design for time-varying optimization problems, which include unconstrained time-varying optimization and a special constrained problem commonly known as a resource allocation problem. The time-varying nature exists in the individual cost functions and the demand functions, and they are then captured by neutrally stable linear dynamic systems known as exosystems. To address the time-varying nature, new distributed algorithm structures are developed and two algorithms are designed for distributed time-varying optimization (DTVO) and distributed time-varying optimal resource allocation (DTVORA) to ensure that there exist time-varying solutions and the solution states will converge to the time-varying solutions. The driving terms for tracking the variation in the solutions are designed using exosystem dynamics. Rigorous convergence analyses are carried out using Lyapunov theory, and the examples are included to demonstrate the potential applications of the two proposed algorithms.

Meta-Learning-Based Adaptive Stability Certificates for Dynamical Systems

This paper addresses the problem of Neural Network (NN) based adaptive stability certification in a dynamical system. The state-of-the-art methods, such as Neural Lyapunov Functions (NLFs), use NN-based formulations to assess the stability of a non-linear dynamical system and compute a Region of Attraction (ROA) in the state space. However, under parametric uncertainty, if the values of system parameters vary over time, the NLF methods fail to adapt to such changes and may lead to conservative stability assessment performance. We circumvent this issue by integrating Model Agnostic Meta-learning (MAML) with NLFs and propose meta-NLFs. In this process, we train a meta-function that adapts to any parametric shifts and updates into an NLF for the system with new test-time parameter values. We demonstrate the stability assessment performance of meta-NLFs on some standard benchmark autonomous dynamical systems.

Asymptotical stabilization of fuzzy semilinear dynamic systems involving the generalized Caputo fractional derivative for $$q \in (1,2)$$

Asymptotical stabilization of fuzzy semilinear dynamic systems involving the generalized Caputo fractional derivative for $$q \in (1,2)$$

Lyapunov stability of fuzzy dynamical systems based on fuzzy-number-valued function granular differentiability

Lyapunov stability theory provides a powerful technique for stability analysis of dynamical systems, particularly for the stability of nonlinear dynamical system. This paper deals with the stability of fuzzy dynamical systems using a novel notion called granular fuzzy Lyapunov function. In order to analyze the stability, some new notions are introduced such as fuzzy equilibrium point, the ball of fuzzy state space, granular fuzzy 2-norm and etc. Based on the concept of fuzzy-number-valued function granular differentiability, two theorems are proved for calculation. Moreover, using granular fuzzy Lyapunov function, the granular fuzzy local stability theorem is proposed, and two definitions of exponential stability and like-Lyapunov stability are generalized. Finally, several examples are given to illustrate the proposed theorems.

A new definition for feature selection stability analysis

AbstractFeature selection (FS) stability is an important topic of recent interest. Finding stable features is important for creating reliable, non-overfitted feature sets, which in turn can be used to generate machine learning models with better accuracy and explanations and are less prone to adversarial attacks. There are currently several definitions of FS stability that are widely used. In this paper, we demonstrate that existing stability metrics fail to quantify certain key elements of many datasets such as resilience to data drift or non-uniformly distributed missing values. To address this shortcoming, we propose a new definition for FS stability inspired by Lyapunov stability in dynamic systems. We show the proposed definition is statistically different from the classical record-stability on ($$n=90$$ n = 90 ) datasets. We present the advantages and disadvantages of using Lyapunov and other stability definitions and demonstrate three scenarios in which each one of the three proposed stability metrics is best suited.

Behavior of solutions of a discrete population model with mutualistic interaction

Abstract We focus on the stability analysis of two types of discrete dynamic models: a discrete dynamic equation and a discrete dynamics system consisting of two equations with mutualistic interaction given by x n + 1 = a + b x n λ − ( x n − 1 + x n − k ) c + x n − 1 + x n − k {x}_{n+1}=a+\frac{b{x}_{n}{\lambda }^{-\left({x}_{n-1}+{x}_{n-k})}}{c+{x}_{n-1}+{x}_{n-k}} and x n + 1 = a 1 + b 1 y n λ − ( y n − 1 + x n − k ) c 1 + y n − 1 + x n − k , y n + 1 = a 2 + b 2 x n λ − ( x n − 1 + y n − k ) c 2 + x n − 1 + y n − k , {x}_{n+1}={a}_{1}+\frac{{b}_{1}{y}_{n}{\lambda }^{-({y}_{n-1}+{x}_{n-k})}}{{c}_{1}+{y}_{n-1}+{x}_{n-k}},\hspace{1.0em}{y}_{n+1}={a}_{2}+\frac{{b}_{2}{x}_{n}{\lambda }^{-\left({x}_{n-1}+{y}_{n-k})}}{{c}_{2}+{x}_{n-1}+{y}_{n-k}}, respectively, where k ∈ { 2 , 3 , … } k\in \left\{2,3,\ldots \right\} , the constants a , a 1 , ≥ 0 a,{a}_{1},\ge 0 , b , b 1 &gt; 0 b,{b}_{1}\gt 0 , and c , c 1 ≥ 0 c,{c}_{1}\ge 0 be the initial densities, finite rate of increase and the limiting constant associated with the density of the species, respectively, and a 2 ≥ 0 {a}_{2}\ge 0 , b 2 &gt; 0 {b}_{2}\gt 0 , and c 2 ≥ 0 {c}_{2}\ge 0 be the initial densities, finite rate of increase, and the limiting constant associated with the density of the mutually interacting species, respectively. k k , a positive integer represents a time delay in the system and λ ≥ 1 \lambda \ge 1 shows a decay factor based on the sum of two past time step population densities. Our main objective is to understand the impact of mutualistic interactions on the stability of discrete dynamic systems. To illustrate the boundedness and stability of these models, we also provide animated plots and bifurcation diagrams.

Fixed time stability of discrete-time stochastic dynamical systems

In this paper, we address fixed time stability in probability of discrete-time stochastic dynamical systems. Unlike finite time stability in probability, wherein the finite time almost sure convergence behavior of the dynamical system depends on the system initial conditions, fixed time stability in probability involves finite time stability in probability for which the stochastic settling-time is guaranteed to be independent of the system initial conditions. More specifically, we develop Lyapunov theorems for fixed time stability in probability for Itô-type stationary nonlinear stochastic difference equations including a Lyapunov theorem that involves a Lyapunov difference satisfying an exponential inequality of the Lyapunov function that gives rise to a minimum bound on the average stochastic settling-time characterized by the primary and secondary branches of the Lambert W function.

On the stability of discrete-time homogeneous polynomial dynamical systems

On the stability of discrete-time homogeneous polynomial dynamical systems

Exponential Stability of Dynamical Systems on Time Scales with Application to Multi-Agent Systems

The exponential stability criteria of systems with time delays on time scales are established, which unifies and generalizes the continuous and discrete cases. The time derivatives of Lyapunov functions (functionals) along solutions are allowed to be indefinite, namely, to take both negative and positive value, which reduces conservatism of the criteria. Moreover, the stability criteria are applicable to both linear and nonlinear systems on time scales, which expands the scope of application of the criteria. Furthermore, the improved stability theorem is applied to solve a leader-following consensus problem of multi-agents on time scales. Sufficient conditions are derived for the leader-following consensus of multi-agent systems under directed interaction topology. A numerical example is given to illustrate the feasibility and effectiveness of the theoretical results.

On the Stability of the Mixing Flow Dynamical System Perturbed with a Logistic Term

This paper continues some recent work in a powerful mathematical domain, with applications in connected scientific fields, namely the stability of dynamical systems. In fluid mechanics, stabilizing a dynamical system is a challenging task and it can be done by various ways. Stabilizing a dynamical system could be often easier if we approach controllable systems, because in this form, there can be imposed some bounds on its behavior, by studying the improvement of the operators that describe the system. In this paper, the mixing flows dynamical systems are taken into account, more exactly the kinematics of mixing flows. The stability analysis of the mixing flow is taken into account, in the case of perturbation with a logistic term. The results can be extended to some other versions of the model.

Investigation of controllability and stability of fractional dynamical systems with delay in control

The primary objective of this research is to investigate the controllability and Hyers–Ulam stability of fractional dynamical systems represented by ψ-Caputo fractional derivative with delay in control. To establish the necessary and sufficient conditions for assessing the controllability of linear fractional systems, which are notably distinguished by the presence of Mittag-Leffler functions, we employ the positivity of the Grammian matrix. Also, we present sufficient conditions for the controllability requirements for nonlinear fractional systems utilizing the fixed-point technique. The Hyers-Ulam stability technique is used to determine the sufficient condition for the stability of fractional nonlinear systems with delay in control. Numerical instances are provided to enhance comprehension of the theoretical findings.