86 research outputs found

    Abstract book

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    Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

    On Asymptotic Stability of Stochastic Differential Equations with Delay in Infinite Dimensional Spaces

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    In most stochastic dynamical systems which describe process in engineering, physics and economics, stochastic components and random noise are often involved. Stochastic effects of these models are often used to capture the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. In the analysis of such systems, we want to investigate their stabilities. This thesis is mainly concerned about the studies of the stability property of stochastic differential equations in infinite dimensional spaces, mainly in Hilbert spaces. Chapter 1 is an overview of the studies. In Chapter 2, we recall basic notations, definitions and preliminaries, especially those on stochastic integration and stochastic differential equations in infinite dimensional spaces. In this way, such notions as Q-Wiener processes, stochastic integrals, mild solutions will be reviewed. We also introduce the concepts of several types of stability. In Chapter 3, we are mainly concerned about the moment exponential stability of neutral impulsive stochastic delay partial differential equations with Poisson jumps. By employing the fixed point theorem, the p-th moment exponential stability of mild solutions to system is obtained. In Chapter 4, we firstly attempt to recall an impulsive-integral inequality by considering impulsive effects in stochastic systems. Then we define an attracting set and study the exponential stability of mild solutions to impulsive neutral stochastic delay partial differential equations with Poisson jumps by employing impulsive-integral inequality. Chapter 5 investigates p-th moment exponential stability and almost sure asymptotic stability of mild solutions to stochastic delay integro-differential equations. Finally in Chapter 6, we study the exponential stability of neutral impulsive stochastic delay partial differential equations driven by a fractional Brownian motion

    Qualitative Theory of Switched Integro-differential Equations with Applications

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    Switched systems, which are a type of hybrid system, evolve according to a mixture of continuous/discrete dynamics and experience abrupt changes based on a switching rule. Many real-world phenomena found in branches of applied math, computer science, and engineering are naturally modelled by hybrid systems. The main focus of the present thesis is on hybrid impulsive systems with distributed delays (HISD). That is, studying the qualitative behaviour of switched integro-differential systems with impulses. Important applications of impulsive systems can be found in stabilizing control (e.g. using impulsive control in combination with switching control) and epidemiology (e.g. pulse vaccination control strategies), both of which are studied in this work. In order to ensure the models are well-posed, some fundamental theory is developed for systems with bounded or unbounded time-delays. Results on existence, uniqueness, and continuation of solutions are established. As solutions of HISD are generally not known explicitly, a stability analysis is performed by extending the current theoretical approaches in the switched systems literature (e.g. Halanay-like inequalities and Razumikhin-type conditions). Since a major field of research in hybrid systems theory involves applying hybrid control to problems, contributions are made by extending current results on stabilization by state-dependent switching and impulsive control for unstable systems of integro-differential equations. The analytic results found are applied to epidemic models with time-varying parameters (e.g. due to changes in host behaviour). In particular, we propose a switched model of Chikungunya disease and study its long-term behaviour in order to develop threshold conditions guaranteeing disease eradication. As a sequel to this, we look at the stability of a more general vector-borne disease model under various vaccination schemes. Epidemic models with general nonlinear incidence rates and age-dependent population mixing are also investigated. Throughout the thesis, computational methods are used to illustrate the theoretical results found

    Stability and Boundedness of Impulsive Systems with Time Delay

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    The stability and boundedness theories are developed for impulsive differential equations with time delay. Definitions, notations and fundamental theory are presented for delay differential systems with both fixed and state-dependent impulses. It is usually more difficult to investigate the qualitative properties of systems with state-dependent impulses since different solutions have different moments of impulses. In this thesis, the stability problems of nontrivial solutions of systems with state-dependent impulses are ``transferred" to those of the trivial solution of systems with fixed impulses by constructing the so-called ``reduced system". Therefore, it is enough to investigate the stability problems of systems with fixed impulses. The exponential stability problem is then discussed for the system with fixed impulses. A variety of stability criteria are obtained and`numerical examples are worked out to illustrate the results, which shows that impulses do contribute to the stabilization of some delay differential equations. To unify various stability concepts and to offer a general framework for the investigation of stability theory, the concept of stability in terms of two measures is introduced and then several stability criteria are developed for impulsive delay differential equations by both the single and multiple Lyapunov functions method. Furthermore, boundedness and periodicity results are discussed for impulsive differential systems with time delay. The Lyapunov-Razumikhin technique, the Lyapunov functional method, differential inequalities, the method of variation of parameters, and the partitioned matrix method are the main tools to obtain these results. Finally, the application of the stability theory to neural networks is presented. In applications, the impulses are considered as either means of impulsive control or perturbations.Sufficient conditions for stability and stabilization of neural networks are obtained

    Bifurcation and Chaos in Fractional-Order Systems

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    This book presents a collection of seven technical papers on fractional-order complex systems, especially chaotic systems with hidden attractors and symmetries, in the research front of the field, which will be beneficial for scientific researchers, graduate students, and technical professionals to study and apply. It is also suitable for teaching lectures and for seminars to use as a reference on related topics

    Fractional Differential Equations, Inclusions and Inequalities with Applications

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    During the last decade, there has been an increased interest in fractional differential equations, inclusions, and inequalities, as they play a fundamental role in the modeling of numerous phenomena, in particular, in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. This book presents collective works published in the recent Special Issue (SI) entitled "Fractional Differential Equation, Inclusions and Inequalities with Applications" of the journal Mathematics. This Special Issue presents recent developments in the theory of fractional differential equations and inequalities. Topics include but are not limited to the existence and uniqueness results for boundary value problems for different types of fractional differential equations, a variety of fractional inequalities, impulsive fractional differential equations, and applications in sciences and engineering

    Nonlinear Dynamics

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    This volume covers a diverse collection of topics dealing with some of the fundamental concepts and applications embodied in the study of nonlinear dynamics. Each of the 15 chapters contained in this compendium generally fit into one of five topical areas: physics applications, nonlinear oscillators, electrical and mechanical systems, biological and behavioral applications or random processes. The authors of these chapters have contributed a stimulating cross section of new results, which provide a fertile spectrum of ideas that will inspire both seasoned researches and students

    Stability of stochastic differential equations in infinite dimensions

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    In engineering, physics and economics, many dynamical systems involving with stochastic components and random noise are often modeled by stochastic models. The stochastic effects of these models are often used to describe the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. Many qualitative properties of such systems have been studied in the past few decades, among which, investigation of stability of such systems is often regarded as the first characteristic of the dynamical systems or models. In general, this thesis is mainly concerned with the studies of the stability property of stochastic differential equations in Hilbert spaces. Chapter 1 is an introduction to a brief history of stochastic differential equations in infinite dimensions, together with an overview of the studies. Chapter 2 is a presentation of preliminaries to some basic stochastic analysis. In Chapter 3, we study the stability in distribution of mild solutions to stochastic delay differential equations with Poisson jumps. Firstly, we use approximation of strong solutions to pass on the stability of strong solutions to the mild ones. Then, by constructing a suitable metric between the transition probability functions of mild solutions, we obtain the desired stability result under some suitable conditions. In Chapter 4, we investigate the stochastic partial delay differential equations with Markovian switching and Poisson jumps. By estimating the coefficients of energy equality, both the exponential stability and almost sure exponential stability of energy solutions to the equations are obtained. In Chapter 5, we study the relationship among strong, weak and mild solutions to the stochastic functional differential equations of neutral type. Finally, in Chapter 6, we study the asymptotic stability of two types of equations, impulsive stochastic delay differential equations with Poisson jumps and stochastic evolution equations with Poisson jumps. By employing the fixed point theorem, we derive the desired stability results under some criteria
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