scholarly journals Existence of Center for Planar Differential Systems with Impulsive Perturbations

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Dengguo Xu
Keyword(s):  
Computing Methods ◽  
Impulsive Systems ◽  
Differential Systems ◽  
Numerical Examples ◽  
State Dependent ◽  
Planar Systems ◽  
Linear And Nonlinear ◽  
Impulsive Perturbations ◽  
The Stability ◽  
Theoretical Results

We present a method that uses successor functions in ordinary differential systems to address the “center-focus” problem of a class of planar systems that have an impulsive perturbation. By deriving solution formulae for impulsive systems, several interesting criteria for distinguishing between the center and the focus of linear and nonlinear planar systems with state-dependent impulsions are established. The conditions describing the stability of the focus of the considered models are also given. The computing methods presented here are more convenient for determining the center of impulsive systems than those in the literature. Numerical examples are given to show the effectiveness of the theoretical results.

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.

Partial stability analysis of nonlinear time-varying impulsive systems

2019 ◽  
Vol 12 (06) ◽  
pp. 1950066
Author(s):  
Boulbaba Ghanmi
Keyword(s):  
Stability Analysis ◽  
Lyapunov Functions ◽  
Time Varying ◽  
Impulsive Systems ◽  
Partial Stability ◽  
Stability Theorems ◽  
Time Varying Systems ◽  
The Stability ◽  
Derivatives Of ◽  
Theoretical Results

This paper investigates the stability analysis with respect to part of the variables of nonlinear time-varying systems with impulse effect. The approach presented is based on the specially introduced piecewise continuous Lyapunov functions. The Lyapunov stability theorems with respect to part of the variables are generalized in the sense that the time derivatives of the Lyapunov functions are allowed to be indefinite. With the help of the notion of stable functions, asymptotic partial stability, exponential partial stability, input-to-state partial stability (ISPS) and integral input-to-state partial stability (iISPS) are considered. Three numerical examples are provided to illustrate the effectiveness of the proposed theoretical results.

Waves, Buckles, and Homoclinics

2009 ◽  
Author(s):  
A. Srikantha Phani
Keyword(s):  
Wave Propagation ◽  
Plane Wave ◽  
Nonlinear Regime ◽  
Lattice Structures ◽  
Linear Regime ◽  
Numerical Examples ◽  
Wave Technique ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Plane Wave Propagation

Dynamic response of lattice structures in linear and nonlinear regime is investigated. In the linear regime, connections between vibration and buckling are revisited in the context of plane wave propagation. The power of wave technique in picking up the correct bifurcation mode and the associated critical load is illustrated. The stability of spatially localised structures arising from homoclinics is examined in the nonlinear regime. Simple analytical results will be presented and illustrated using numerical examples.

scholarly journals Analysis of a Class of Fractional Nonlinear Multidelay Differential Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Zhuoyan Gao ◽  
JinRong Wang ◽  
Yong Zhou
Keyword(s):  
Iterative Learning Control ◽  
Time Delays ◽  
Iterative Learning ◽  
Multiple Time ◽  
Weighted Norm ◽  
Differential Systems ◽  
Numerical Examples ◽  
Multiple Time Delays ◽  
Two Parameters ◽  
Theoretical Results

We address existence and Ulam-Hyers and Ulam-Hyers-Mittag-Leffler stability of fractional nonlinear multiple time-delays systems with respect to two parameters’ weighted norm, which provides a foundation to study iterative learning control problem for this system. Secondly, we design PID-type learning laws to generate sequences of output trajectories to tracking the desired trajectory. Two numerical examples are used to illustrate the theoretical results.

scholarly journals Mean Square Stability of Impulsive Stochastic Differential Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Shujie Yang ◽  
Bao Shi ◽  
Mo Li
Keyword(s):  
Stochastic Analysis ◽  
Functional Method ◽  
Stochastic Equations ◽  
Mean Square ◽  
Differential Systems ◽  
Numerical Examples ◽  
Mean Square Stability ◽  
Analysis Theory ◽  
Delay Dependent ◽  
Theoretical Results

Based on Lyapunov-Krasovskii functional method and stochastic analysis theory, we obtain some new delay-dependent criteria ensuring mean square stability of a class of impulsive stochastic equations. Numerical examples are given to illustrate the effectiveness of the theoretical results.

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 QUALITATIVE BEHAVIOURS OF A SYSTEM OF NONLINEAR DIFFERENCE EQUATIONS

2021 ◽  
Vol 21 (1) ◽  
pp. 39-56
Author(s):  
ERKAN TAŞDEMİR ◽  
YÜKSEL SOYKAN
Keyword(s):  
Real Number ◽  
Difference Equations ◽  
Equilibrium Points ◽  
Bounded Solutions ◽  
Nonlinear Difference Equations ◽  
Related System ◽  
Numerical Examples ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Theoretical Results

The paper aims to study the dynamics of a system of nonlinear difference equations x_(n+1)=x_(n-1) y_n+A,y_(n+1)=y_(n-1) x_n+A where A is real number. We especially investigate the stability of equilibrium points, convergence of equilibrium points, existence of periodic solutions, and existence of bounded solutions of related system. Moreover, we present some numerical examples to verify the theoretical results.

CHAOTIFICATION OF LINEAR IMPULSIVE DIFFERENTIAL SYSTEMS WITH APPLICATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250297 ◽  
Author(s):  
QIGUI YANG ◽  
GUIRONG JIANG ◽  
TIANSHOU ZHOU
Keyword(s):  
Dynamical Systems ◽  
Research Field ◽  
Canonical Forms ◽  
Impulsive Systems ◽  
Differential Systems ◽  
Practical Applications ◽  
Discrete Maps ◽  
Impulsive Differential Systems ◽  
Technical Details ◽  
Theoretical Results

Chaotification of dynamical systems is a hot topic in the research field of chaos in recent years. Previous studies showed that (even linear) discrete or continuous dynamical systems can be chaotified by designing appropriate controllers. Here, we study chaotification of linear impulsive differential systems. First, we propose a framework for chaotification of general linear impulsive differential systems that can be transformed into discrete maps. Then, we give technical details for how to chaotify several typical linear impulsive differential systems that are actually canonical forms, including how to design appropriate quadratic impulsive controllers, how to find snapback repellers in the Marotto theorem, etc. As one of the main theoretical results, we rigorously prove the existence of chaos in all the considered impulsive systems. In addition, numerical examples are used to verify the theoretical prediction in each case. We are expecting that our proposed approach can have practical applications in the engineering field.

scholarly journals Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Hai Zhang ◽  
Daiyong Wu ◽  
Jinde Cao ◽  
Hui Zhang
Keyword(s):  
Asymptotic Stability ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Algebraic Approach ◽  
Differential Systems ◽  
Order Linear ◽  
Delay Differential Systems ◽  
The Stability ◽  
Delay Differential ◽  
Theoretical Results

We investigate the delay-independently asymptotic stability of fractional-order linear singular delay differential systems. Based on the algebraic approach, the sufficient conditions are presented to ensure the asymptotic stability for any delay parameter. By applying the stability criteria, one can avoid solving the roots of transcendental equations. An example is also provided to illustrate the effectiveness and applicability of the theoretical results.

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