scholarly journals Identifying the stability of the equilibrium position using adaptive regulator with the help of the second (direct) Lyapunov method

2018 ◽  
Author(s):  
A .G. Chuikova
Keyword(s):  
Equilibrium Position ◽  
Lyapunov Method ◽  
Direct Lyapunov Method ◽  
The Stability ◽  
Adaptive Regulator

scholarly journals Stability with Respect to a Part of Variables under Constant Perturbations of the Partial Equilibrium Position of Differential Equation Nonlinear Systems

2018 ◽  
Vol 28 (3) ◽  
pp. 344-351
Author(s):  
Pavel P. Lipasov ◽  
Vladimir N. Shchennikov
Keyword(s):  
Nonlinear Systems ◽  
Equilibrium Position ◽  
Lyapunov Method ◽  
Stability Theorem ◽  
Partial Equilibrium ◽  
Dynamic Processes ◽  
Stability Theorems ◽  
Research Objects ◽  
Controlled Motion ◽  
The Stability

Introduction. It is impossible to take into account all the forces acting in the process of mathematical modeling of dynamic processes. In order that mathematical models the most accurately describe the dynamic processes, they must include the terms that correspond the constant perturbations. These problems arise in applied tasks. In this paper we consider the case when the system allows for the partial equilibrium position. The aim of this work is to prove the stability theorem for the partial equilibrium position at constant perturbations, which are small at every instant. Materials and Methods. The research objects are nonlinear systems of differential equations that allow for a partial equilibrium position. Using the second Lyapunov method, there are proved the stability theorems for the constant perturbations of the partial equilibrium position, which are small at every instant. Results. Together with the introduction of stability for a part of the variables, it has become necessary to introduce stability for the part of phase variables under constant perturbations. The first stability theorem of the part of phase variables under constant perturbations was obtained by A. S. Oziraner. In this work, we prove a theorem of the stability of the constant perturbations of the partial equilibrium position, small at every instant. It should be noted that there is no stability theorems of constant perturbations for the partial equilibrium position. Thus, the theorem proved in this work is of a pioneer nature. Conclusions. The theorem 3 proved in the work is the development of the mathematical theory of stability. The results of this work are applicable in the mechanics of controlled motion, nonlinear system.

scholarly journals On the stability of solutions of fractional non conformable differential equations

2020 ◽  
Vol 65 (4) ◽  
pp. 495-502
Author(s):  
Paulo M. Guzman ◽  
Luciano M. Lugo Motta Bittencurt ◽  
Juan E. Napoles Valdes
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Method ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Direct Lyapunov Method ◽  
Stability Of Solutions ◽  
Fractional Differential ◽  
The Stability

In this note we obtain sufficient conditions under which we can guarantee the stability of solutions of a fractional differential equations of non conformable type and we obtain some fractional analogous theorems of the direct Lyapunov method for a given class of equations of motion.

scholarly journals Analysis of the most common methods for determining the stability of energy systems

2021 ◽  
pp. 17-26
Author(s):  
Andrii Hnatov ◽  
Shchasiana Arhun ◽  
Ruslan Bagach ◽  
Hanna Hnatova ◽  
Valentina Tarasova ◽  
...  
Keyword(s):  
Power Systems ◽  
Lyapunov Method ◽  
Direct Method ◽  
Energy Systems ◽  
Energy System ◽  
Mathematical Function ◽  
Direct Lyapunov Method ◽  
First Approximation ◽  
Differential Methods ◽  
The Stability

Problem. There are many methods for determining the stability of the energy system. In normal operating condition (normal rated mode), the power system must reliably ensure the consumption of electricity of normalized quality. However, in addition to the normal state, there are emergency and transient states caused by various transients. This is due to the fact that the energy system is constantly changing its parameters. Such changes are determined by variations in the amount of power produced and consumed, as well as the changes in system configuration. Goal. The goal is studying the possibilities of various methods of determining the power systems stability and drawing up the general algorithm of actions for maintenance of their stability. Methodology. When determining the stability of energy systems by the Lyapunov method, two methods can be used: the direct method and the first approximation method. Lyapunov direct method refers to differential methods. To conclude about the stability of the system we do not find a general or particular solution of differential equations, but with their help we find a mathematical function, the complete derivative of which over time allows to obtain a conclusion about the stability of the system. Results. Many methods can be used to determine whether a sustainable energy system is stable or not. The most common are the Lyapunov methods and the Moiseev method. It is determined that the direct Lyapunov method refers to differential methods. The application of the direct Lyapunov method for energy problems is limited. Currently, it can be used only for some individual cases. The method of the first approximation (Lyapunov first method) has received wider application in the solution of power problems. When applying this method, which belongs to the group of methods of full integration, the right-hand sides of the equations are decomposed into power series. Originality. It is determined that one of the perspective directions of increasing the efficiency of the mathematical device work is using the methods of the second order in modeling and optimization of operating modes of electric power systems. This allows you to increase the speed and reliability of the convergence of iterative processes. Practical value. Based on the analysis of various existing methods for solving the problems of stability of energy systems, an algorithm of actions is proposed and developed, which will help to solve the problem of stability in practice.

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

2020 ◽  
Vol 21 (7-8) ◽  
pp. 749-759
Author(s):  
Rachida Mezhoud ◽  
Khaled Saoudi ◽  
Abderrahmane Zaraï ◽  
Salem Abdelmalek
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Linearization Method ◽  
Direct Lyapunov Method ◽  
Fractional Systems ◽  
Reaction Diffusion Model ◽  
Theoretical Results

AbstractFractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

Global stability analysis of fractional-order gene regulatory networks with time delay

2019 ◽  
Vol 12 (06) ◽  
pp. 1950067 ◽  
Author(s):  
Zhaohua Wu ◽  
Zhiming Wang ◽  
Tiejun Zhou
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Gene Regulatory Networks ◽  
Existence And Uniqueness ◽  
Regulatory Networks ◽  
Lyapunov Method ◽  
Regulation Mechanism ◽  
Global Stability Analysis ◽  
Gene Regulatory ◽  
The Stability

Fractional-order gene regulatory networks with time delay (DFGRNs) have proven that they are more suitable to model gene regulation mechanism than integer-order. In this paper, a novel DFGRN is proposed. The existence and uniqueness of the equilibrium point for the DFGRN are proved under certain conditions. On this basis, the conditions on the global asymptotic stability are established by using the Lyapunov method and comparison theorem for the DFGRN, and the stability conditions are dependent on the fractional-order [Formula: see text]. Finally, numerical simulations show that the obtained results are reasonable.

On the stability of a mathematical model for HIV(AIDS) - cancer dynamics

2021 ◽  
Vol 8 (4) ◽  
pp. 783-796
Author(s):  
H. W. Salih ◽  
◽  
A. Nachaoui ◽  
Keyword(s):  
Mathematical Model ◽  
Highly Active Antiretroviral Therapy ◽  
Lyapunov Method ◽  
Equilibrium Points ◽  
Hiv Treatment ◽  
Biochemical Processes ◽  
Highly Active ◽  
Biological Phenomena ◽  
The Stability ◽  
The Impact

In this work, we study an impulsive mathematical model proposed by Chavez et al. [1] to describe the dynamics of cancer growth and HIV infection, when chemotherapy and HIV treatment are combined. To better understand these complex biological phenomena, we study the stability of equilibrium points. To do this, we construct an appropriate Lyapunov function for the first equilibrium point while the indirect Lyapunov method is used for the second one. None of the equilibrium points obtained allow us to study the stability of the chemotherapeutic dynamics, we then propose a bifurcation of the model and make a study of the bifurcated system which contributes to a better understanding of the underlying biochemical processes which govern this highly active antiretroviral therapy. This shows that this mathematical model is sufficiently realistic to formulate the impact of this treatment.

Asymptotic Behavior of Periodic Cohen-Grossberg Neural Networks with Delays

2009 ◽  
Vol 21 (12) ◽  
pp. 3444-3459 ◽  
Author(s):  
Wei Lin
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Lyapunov Method ◽  
Autonomous Systems ◽  
Neural Systems ◽  
Stability Criteria ◽  
Volterra System ◽  
Numerical Examples ◽  
The Stability ◽  
Special Case

Without assuming the positivity of the amplification functions, we prove some M-matrix criteria for the [Formula: see text]-global asymptotic stability of periodic Cohen-Grossberg neural networks with delays. By an extension of the Lyapunov method, we are able to include neural systems with multiple nonnegative periodic solutions and nonexponential convergence rate in our model and also include the Lotka-Volterra system, an important prototype of competitive neural networks, as a special case. The stability criteria for autonomous systems then follow as a corollary. Two numerical examples are provided to show that the limiting equilibrium or periodic solution need not be positive.

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