scholarly journals The Stability of Nonlinear Differential Systems with Random Parameters

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Josef Diblík ◽  
Irada Dzhalladova ◽  
Miroslava Růžičková
Keyword(s):  
Trivial Solution ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
New Method ◽  
Random Parameters ◽  
Differential Systems ◽  
The Stability ◽  
Nonlinear Differential Systems

The paper deals with nonlinear differential systems with random parameters in a general form. A new method for construction of the Lyapunov functions is proposed and is used to obtain sufficient conditions forL2-stability of the trivial solution of the considered systems.

scholarly journals Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
D. Ya. Khusainov ◽  
J. Diblík ◽  
Z. Svoboda ◽  
Z. Šmarda
Keyword(s):  
Trivial Solution ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Dimensional System ◽  
Global Instability ◽  
Degree Polynomial ◽  
Differential Systems ◽  
Right Hand ◽  
The Matrix ◽  
Full Derivative

The present investigation deals with global instability of a generaln-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.

scholarly journals Asymptotic Stability of Caputo Type Fractional Neutral Dynamical Systems with Multiple Discrete Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hai Zhang ◽  
Daiyong Wu ◽  
Jinde Cao
Keyword(s):  
Asymptotic Stability ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Algebraic Approach ◽  
Stability Criteria ◽  
Differential Systems ◽  
Discrete Delays ◽  
Transcendental Equations ◽  
The Stability ◽  
Theoretical Results

We discuss the delay-independent asymptotic stability of Caputo type fractional-order neutral differential systems with multiple discrete delays. Based on the algebraic approach and matrix theory, the sufficient conditions are derived to ensure the asymptotic stability for all time-delay parameters. By applying the stability criteria, one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and convenient. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed theoretical results.

scholarly journals Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Fengrong Zhang ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Caputo Derivative ◽  
Sufficient Conditions ◽  
Comparison Method ◽  
Asymptotical Stability ◽  
Sufficient Condition ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
The Stability

This paper deals with the stability of nonlinear fractional differential systems equipped with the Caputo derivative. At first, a sufficient condition on asymptotical stability is established by using a Lyapunov-like function. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In addition, some other sufficient conditions on stability are also presented.

scholarly journals Common Weak Linear Copositive Lyapunov Functions for Positive Switched Linear Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Yuangong Sun ◽  
Zhaorong Wu ◽  
Fanwei Meng
Keyword(s):  
Stability Analysis ◽  
Complex Systems ◽  
Linear Systems ◽  
Algebraic Structure ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Switched Linear Systems ◽  
Numerical Examples ◽  
The Stability ◽  
Necessary And Sufficient

Lyapunov functions play a key role in the stability analysis of complex systems. In this paper, we study the existence of a class of common weak linear copositive Lyapunov functions (CWCLFs) for positive switched linear systems (PSLSs) which generalize the conventional common linear copositive Lyapunov functions (CLCLFs) and can be used as handy tool to deal with the stability of PSLSs not covered by CLCLFs. We not only establish necessary and sufficient conditions for the existence of CWCLFs but also clearly describe the algebraic structure of all CWCLFs. Numerical examples are also given to demonstrate the effectiveness of the obtained results.

scholarly journals Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 744 ◽  
Author(s):  
Bei Zhang ◽  
Yonghui Xia ◽  
Lijuan Zhu ◽  
Haidong Liu ◽  
Longfei Gu
Keyword(s):  
Dynamical System ◽  
Graph Theory ◽  
Global Stability ◽  
Fractional Order ◽  
Trivial Solution ◽  
Sufficient Conditions ◽  
Fractional Order System ◽  
Coupled Systems ◽  
Order System ◽  
The Stability

Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper.

scholarly journals Integral Stability in Terms of Two Measures for Nonlinear Differential Systems with “Maxima”

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Peiguang Wang ◽  
Qing Xu ◽  
Xiaojing Liu
Keyword(s):  
Comparison Principle ◽  
Lyapunov Functions ◽  
Differential Systems ◽  
Two Measures ◽  
Integral Stability ◽  
Nonlinear Differential Systems

This paper investigates relatively integral stability in terms of two measures for two differential systems with “maxima” by employing Lyapunov functions, Razumikhin method, and comparison principle. An example is given to illustrate our result.

scholarly journals A New Method for Research on the Center-Focus Problem of Differential Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Zhengxin Zhou
Keyword(s):  
Critical Point ◽  
Sufficient Conditions ◽  
New Method ◽  
Qualitative Behavior ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Planar Polynomial Differential Systems

We will introduce Mironenko’s method to discuss the Poincaré center-focus problem, and compare the methods of Lyapunov and Mironenko. We apply the Mironenko method to discuss the qualitative behavior of solutions of some planar polynomial differential systems and derive the sufficient conditions for a critical point to be a center.

scholarly journals A comparison principle and stability for large-scale impulsive delay differential systems

2005 ◽  
Vol 47 (2) ◽  
pp. 203-235 ◽  
Author(s):  
Xinzhi Liu ◽  
Xuemin Shen ◽  
Yi Zhang
Keyword(s):  
Comparison Principle ◽  
Large Scale ◽  
Sufficient Conditions ◽  
Neutral Systems ◽  
Differential Systems ◽  
Impulsive Delay ◽  
Linear And Nonlinear ◽  
Delay Differential Systems ◽  
The Stability ◽  
Delay Differential

AbstractThis paper studies the stability of large-scale impulsive delay differential systems and impulsive neutral systems. By developing some impulsive delay differential inequalities and a comparison principle, sufficient conditions are derived for the stability of both linear and nonlinear large-scale impulsive delay differential systems and impulsive neutral systems. Examples are given to illustrate the main results.

scholarly journals h-Stability of Linear Matrix Differential Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Peiguang Wang ◽  
Xiaowei Liu
Keyword(s):  
Stability Problem ◽  
Kronecker Product ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix System ◽  
Differential Systems ◽  
Perturbed System ◽  
Matrix Differential ◽  
The Stability ◽  
Product Of Matrices

This paper investigates the stability problem of linear matrix differential systems and gives some sufficient conditions ofh-stability for linear matrix system and its associated perturbed system by using the Kronecker product of matrices. An example is also worked out to illustrate our results.

scholarly journals Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 322
Author(s):  
Ricardo Almeida ◽  
Ravi P. Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Lyapunov Functions ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Hopfield Neural Network ◽  
Quadratic Lyapunov Functions ◽  
Fractional Differential ◽  
The Stability ◽  
Theoretical Results

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

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