scholarly journals Unpredictable Solutions of Linear Impulsive Systems

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1798
Author(s):  
Marat Akhmet ◽  
Madina Tleubergenova ◽  
Mehmet Onur Fen ◽  
Zakhira Nugayeva
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Random Process ◽  
Logistic Map ◽  
Impulsive Differential Equations ◽  
Impulsive Systems ◽  
New Type ◽  
Piecewise Continuous ◽  
Definition Of ◽  
Theoretical Results

We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provided. We benefit from the B-topology in the space of discontinuous functions on the purpose of proving the presence of unpredictable solutions. For constructive definitions of unpredictable components in examples, randomly determined unpredictable sequences are newly utilized. Namely, the construction of a discontinuous unpredictable function is based on an unpredictable sequence determined by a discrete random process, and the set of discontinuity moments is realized by the logistic map. Examples with numerical simulations are presented to illustrate the theoretical results.

scholarly journals Dynamic Behaviors of an SEIR Epidemic Model in a Periodic Environment with Impulse Vaccination

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mei Yan ◽  
Zhongyi Xiang
Keyword(s):  
Differential Equations ◽  
Disease Control ◽  
Numerical Simulations ◽  
Epidemic Model ◽  
Floquet Theory ◽  
Impulsive Differential Equations ◽  
Threshold Parameter ◽  
Pulse Vaccination ◽  
Periodic Environment ◽  
Theoretical Results

We consider a nonautonomous SEIR endemic model with saturation incidence concerning pulse vaccination. By applying Floquet theory and the comparison theorem of impulsive differential equations, a threshold parameter which determines the extinction or persistence of the disease is presented. Finally, numerical simulations are given to illustrate the main theoretical results and it shows that pulse vaccination plays a key role in the disease control.

scholarly journals Li–Yorke Chaos in Hybrid Systems on a Time Scale

2015 ◽  
Vol 25 (14) ◽  
pp. 1540024 ◽  
Author(s):  
Marat Akhmet ◽  
Mehmet Onur Fen
Keyword(s):  
Differential Equations ◽  
Hybrid Systems ◽  
Time Scales ◽  
Time Scale ◽  
Impulsive Differential Equations ◽  
Reduction Technique ◽  
Duffing Equation ◽  
Dynamic Equations ◽  
Definition Of ◽  
First Time

By using the reduction technique to impulsive differential equations [Akhmet & Turan, 2006], we rigorously prove the presence of chaos in dynamic equations on time scales (DETS). The results of the present study are based on the Li–Yorke definition of chaos. This is the first time in the literature that chaos is obtained for DETS. An illustrative example is presented by means of a Duffing equation on a time scale.

scholarly journals On the General Solution of Impulsive Systems with Hadamard Fractional Derivatives

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Xianmin Zhang ◽  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Wenbin Ding ◽  
Hui Cao ◽  
...  
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Theoretical Result ◽  
Fractional Derivatives ◽  
Impulsive Differential Equations ◽  
Limit Case ◽  
Impulsive Systems ◽  
Fractional System ◽  
Approach Zero

This paper is concerned with the solution for impulsive differential equations with Hadamard fractional derivatives. The general solution of this impulsive fractional system is found by considering the limit case in which impulses approach zero. Next, an example is provided to expound the theoretical result.

scholarly journals EVENTUAL STABILITY AND EVENTUAL BOUNDEDNESS FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH “SUPREMUM”

2011 ◽  
Vol 16 (1) ◽  
pp. 304-314 ◽  
Author(s):  
Ivanka Stamova
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lyapunov Functions ◽  
Impulsive Differential Equations ◽  
Comparison Method ◽  
Applied Method ◽  
Piecewise Continuous ◽  
Eventual Stability

Eventual stability and eventual boundedness for nonlinear impulsive differential equations with supremums are studied. The impulses take place at fixed moments of time. Piecewise continuous Lyapunov functions have been applied. Method of Razumikhin as well as comparison method for scalar impulsive ordinary differential equations have been employed.

scholarly journals Adaptive Synchronization of Complex Dynamical Multilinks Networks with Similar Nodes

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Weiping Wang ◽  
Lixiang Li ◽  
Haipeng Peng ◽  
Jialiang Yuan ◽  
Jinghua Xiao ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Stability ◽  
Real World ◽  
Stability Theory ◽  
Lyapunov Stability Theory ◽  
Complex Dynamical Networks ◽  
Adaptive Synchronization ◽  
Definition Of ◽  
First Time ◽  
Theoretical Results

This paper studies the synchronization of complex dynamical networks with multilinks and similar nodes. The dynamics of all the nodes in the networks are impossible to be completely identical due to the differences of parameters or the existence of perturbations. Networks with similar nodes are universal in the real world. In order to depict the similarity of the similar nodes, we give the definition of the minimal similarity of the nodes in the network for the first time. We find the threshold of the minimal similarity of the nodes in the network. If the minimal similarity of the nodes is bigger than the threshold, then the similar nodes can achieve synchronization without controllers. Otherwise, adaptive synchronization method is adopted to synchronize similar nodes in the network. Some new synchronization criteria are proposed based on the Lyapunov stability theory. Finally, numerical simulations are given to illustrate the feasibility and the effectiveness of the proposed theoretical results.

scholarly journals Uniqueness of Solutions, Stability and Simulations for a Differential Problem Involving Convergent Series and Time Variable Singularities

2021 ◽  
Author(s):  
Yazid GOUARI ◽  
Zoubir Dahmani ◽  
Meriem Mansouria BELHAMITI ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Convergent Series ◽  
Time Variable ◽  
Kutta Method ◽  
Integral Conditions ◽  
Uniqueness Of Solutions ◽  
Differential Problem ◽  
New Type ◽  
Nonlocal Integral Conditions

We focus on a new type of nonlinear integro-differential equations with nonlocal integral conditions. The considered problem has one nonlinearity with time variable singularity. It involves also some convergent series combined to Riemann-Liouville integrals. We prove a uniqueness of solutions for the proposed problem, then, we provide some examples to illustrate this result. Also, we discuss the Ulam-Hyers stability for the problem. Some numerical simulations, using Rung Kutta method, are discussed too. At the end, a conclusion follows.

scholarly journals Stability Results for a Coupled System of Impulsive Fractional Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 927 ◽  
Author(s):  
Akbar Zada ◽  
Shaheen Fatima ◽  
Zeeshan Ali ◽  
Jiafa Xu ◽  
Yujun Cui
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Coupled System ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Stability Results ◽  
Theoretical Results

In this paper, we establish sufficient conditions for the existence, uniqueness and Ulam–Hyers stability of the solutions of a coupled system of nonlinear fractional impulsive differential equations. The existence and uniqueness results are carried out via Banach contraction principle and Schauder’s fixed point theorem. The main theoretical results are well illustrated with the help of an example.

scholarly journals The stochastic θ method for stationary distribution of stochastic differential equations with Markovian switching

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yanan Jiang ◽  
Liangjian Hu ◽  
Jianqiu Lu
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Stochastic Differential Equations ◽  
Stationary Distribution ◽  
Existence And Uniqueness ◽  
Numerical Solutions ◽  
Markovian Switching ◽  
Theoretical Results

AbstractIn this paper, stationary distribution of stochastic differential equations (SDEs) with Markovian switching is approximated by numerical solutions generated by the stochastic θ method. We prove the existence and uniqueness of stationary distribution of the numerical solutions firstly. Then, the convergence of numerical stationary distribution to the underlying one is discussed. Numerical simulations are conducted to support the theoretical results.

scholarly journals On Nonlinear Forced Impulsive Differential Equations under Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2066
Author(s):  
Shyam Sundar Santra ◽  
Hammad Alotaibi ◽  
Samad Noeiaghdam ◽  
Denis Sidorov
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Impulsive System ◽  
Bounded Solutions ◽  
Oscillatory Behaviour ◽  
Impulsive Systems ◽  
Forcing Term

This study is connected with the nonoscillatory and oscillatory behaviour to the solutions of nonlinear neutral impulsive systems with forcing term which is studied for various ranges of of the neutral coefficient. Furthermore, sufficient conditions are obtained for the existence of positive bounded solutions of the impulsive system. The mentioned example shows the feasibility and efficiency of the main results.

scholarly journals Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1535
Author(s):  
Marat Akhmet ◽  
Madina Tleubergenova ◽  
Akylbek Zhamanshin
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Existence And Uniqueness ◽  
Asymptotically Stable ◽  
Stable Solutions ◽  
Theoretical Results

In this paper, modulo periodic Poisson stable functions have been newly introduced. Quasilinear differential equations with modulo periodic Poisson stable coefficients are under investigation. The existence and uniqueness of asymptotically stable modulo periodic Poisson stable solutions have been proved. Numerical simulations, which illustrate the theoretical results are provided.

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