A novel approach for complete identification of dynamic. Dynamical systems theory also known as dynamic systems theory or just systems theory is a series of principles and tools for studying change. Modeling, analysis and simulation presents a study of fractionalorder chaotic systems accompanied by matlab programs for simulating their state space trajectories, which are shown in the illustrations in the book. Also, some examples for fractional dynamical systems of this type are given. The fractional dynamic process of a quantum system is a novel and complicated problem. However, the fractional dynamic system 2 is stable as 0 0 changes the fractional dynamic systems which needs to be considered depending on the. In this regard, it is necessary and important to study the controllability problems for fractional singular dynamical systems. The examples of the fractional standard and logistic maps demonstrate that phase space of nonlinear fractional dynamical systems may. Fractional dynamical system and its linearization theorem. Consensus of a class of nonlinear fractionalorder multi. The book is devoted to recent developments in the theory of fractional calculus and its. This book describes theoretical and numerical aspects. Basic theory of dynamical systems a simple example.
In this work we address the dynamic simulation and optimization of chemical processing systems modeled in terms of fractional order differential equations. In the fields of dynamical systems and control theory, a fractional order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of noninteger order. System identification, which refers to estimation of process parameters, is a necessity in control theory. Number theory and dynamical systems 4 some dynamical terminology a point. In section 2, we present some essential facts in fractional calculus and stochastic analysis that will be used to obtain our main results. In order to achieve this goal, it is first necessary to develop the local theory for fractional nonlinear systems. Lakshmikantham and others published theory of fractional dynamic systems find, read and cite all the research you need on researchgate.
This invaluable book is devoted to a rapidly developing area on the research of the qualitative theory of fractional differential equations. In this paper, we studied the stabilization of nonlinear fractional order dynamic systems. The concept offers an explanation for the physical mechanism of variableorder fractional dynamic systems and multi system interaction. The results are obtained by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the schauder fixed point theorem. In the general case, the fractional hamiltonian or gradient systems cannot be considered as hamiltonian gradient systems. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first.
This book features original research articles on the topic of mathematical modelling and fractional differential equations. It is selfcontained and unified in presentation, and provides readers the necessary background material required to go further into the subject and explore. An excellent account of the study of fractional dierential equations can be found in 2,3,19,34, 39. Based on the qualitative theory, the existence and uniqueness of solution for the fractional order. Stochastic averaging principle for dynamical systems with. The contributors are selected based on their active and important contributions to their respective topics. Systems thinkers consider the dynamic interactions of all the factors contributing to a particular trait of interest. Sliding mode control for generalized robust synchronization of mismatched fractional order dynamical systems and its application to secure transmission of voice messages p. Lakshmikantham and others published theory of fractional dynamic systems find, read and cite all the research you need. This study presents an inequality which can be used to analyse the stability of fractional order systems by constructing lyapunov functions. Accurate modeling of many dynamic systems leads to a set of fractional differential equations fdes. In the fields of dynamical systems and control theory, a fractionalorder system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of noninteger order. Fractional derivatives, fractional tangent bundle, frac. This book aims to introduce some new trends and results on the study of the fractional differential equations, and to provide a good understanding of this field to beginners who are interested in this field, which is the authors beautiful hope.
We proposed the definition of mittagleffler stability and the fractional lyapunov direct method, which enriched the knowledge of. The solar system sun and planets, the weather, the motion of billiard balls on a billiard table, sugar dissolving in a cup of coffee, the growth of crystals. Advances in lyapunov theory of caputo fractionalorder systems. Dynamic systems theory dst is a broad theoretical framework imported from the physical sciences and used in psychology and cognitive science in the past several decades that provides an alternative to the computational and informationprocessing approach that has. Therefore, it is important to study the qualitative behavior of fractional dynamical systems. Fractional integrals and derivatives theory and applications. New methods in theory and cosmic applications of spherical random fields.
About robust stability of caputo linear fractional dynamic systems with time delays through fixed point theory, fixed point theory appl. These notes are devoted to some methods used in the fractional calculus theory of integration and differen tiation of an arbitrary order and to application of the. While fractional derivatives were first proposed by liouville in 1832 samko et al. Dynamical processes and dynamical systems of fractional order attract. Fractional dynamics world scientific publishing company. Their work laid the foundation for a fresh approach to understanding how people learn, grow, and change. As a result, the change of initial value in time from zero to any t0 0 changes the fractional dynamic systems which needs to be considered depending on the. Fractional order control a tutorial ieee conference. Paper t itle d monotonicity, concavity, and convexity of fractional derivative of functions by x. This paper presents a general formulation and a solution scheme for a class of fractional optimal control problems focps for those systems. In this section, the maxent principle together with fractional moments are applied to evaluate the eevd for reliability assessment of structural dynamic systems. Theory of fractional dynamic systems request pdf researchgate.
This site is like a library, use search box in the widget to get ebook that you want. Conclusion and future work in this paper, we studied the stabilization of nonlinear fractional order dynamic systems. Formally, dynamic systems theory is an abstract framework, based on concepts from thermodynamics and nonlinear mathematics. One of the basic qualitative behaviours of a dynamical system is stability. Pdf in this epj special topics issue selected contributions on some recent developments in the theory and applications of fractional. This is the first book that attempts to describe the theory of fractional differential equations as an independent discipline. Reachability and controllability of fractional singular. Derivatives and integrals of fractional orders are used to describe objects that can be characterized by powerlaw nonlocality, powerlaw longrange dependence or fractal properties. The performance index of a focp is considered as a function of both. Real processes are usually of fractional order as opposed to the ideal integral order models. This volume provides the latest developments in the field of fractional dynamics, which covers fractional anomalous transport phenomena, fractional statistical mechanics, fractional quantum mechanics and fractional quantum field theory. In this paper, we have studied the stability of a class of nonlinear fractional neutral differential difference systems. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future.
But we can still establish the similar relationship between a fractional differential equation and. We will have much more to say about examples of this sort later on. Applications of fractional calculus semantic scholar. This book reports on an outstanding research devoted to modeling and control of dynamic systems using fractional order calculus.
It describes the development of modelbased control design methods for systems described by fractional dynamic models. Stability of caputo fractional differential equations by. By using the mittagleffler function, laplace transform and the gronwallbellman lemma, one sufficient condition is attained for the asymptotical stability of a class of nonlinear fractional differential systems whose order lies in 0, 2. Fractional calculus can more accurately describe the existence characteristics of complex physical, financial or medical systems, and can truly reflect the actual state properties of these systems. Numerical solutions are used to verify the analytical results.
The fractional dynamics of quantum systems sciencedirect. The purpose of this course is to introduce some basic concepts and tools which are useful in the analysis and control of dynamical systems. Observability of nonlinear fractional dynamical systems. Stability of fractional neutral systems advances in. We introduce the notion of hstability for fractional differential systems. The examples of the fractional standard and logistic maps demonstrate that phase space of nonlinear fractional dynamical systems may contain periodic. Pdf stabilizability of fractional dynamical systems. We introduced the fractional comparison principle for caputo fractional order systems, which enriched the knowledge of both system theory and fractional calculus. Dynamic analysis of a fractional order r ossler system.
The investigation is performed via fixed point theory in a complete metric space by defining appropriate nonexpansive or contractive selfmappings from initial. Jul 29, 2019 immediate online access to all issues from 2019. A new method for reliability assessment of structural dynamic. Dynamic systems theory in social psychology iresearchnet. Roughly speaking, the theory of fractional averaging provides us ample opportunity to accurately reveal the essence of real life14. Stability and stabilization for a class of semilinear. Fractional averaging principle is in the innovation phase. Fractional dynamics applications of fractional calculus. The advantage of using fractional moments is that the distribution of a positive random variable can be characterized by a relatively few number of fractional moments. Fractional order control systems download ebook pdf, epub. In this paper we establish a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. Fractional order control systems download ebook pdf. Now there are so many books about the fractional calculus, but i think the book, theory and applications of fractional differential equations, volume 204 northholland mathematics studies is an excellent handbook for those people who major in fractional calculus, fractional differential equations, especially in the theory research. Chaos theory is a branch of mathematics focusing on the study of chaosstates of dynamical systems whose apparentlyrandom states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions.
For now, we can think of a as simply the acceleration. On fractional derivatives fractionalorder dynamic systems and pi. The investigation is performed via fixed point theory in a complete metric space by defining appropriate nonexpansive or contractive self mappings from. Dynamic systems theory permits us to understand how cultural difference becomes bodily difference.
Recent trends in theory and applications published on tuesday, 04 april 2017 18. Fractional dynamical systems international journal of nonlinear. Description of the chaotic systems is clearly presented. Fractional dynamic models for mri to probe tissue microstructure. Fractionalorder modeling and control of dynamic systems. The suggested class of fractional gradient and hamiltonian systems is wider tarasov, 2005a,b than the usual class of gradient and hamiltonian dynamical systems. Dynamical systems theory attempts to understand, or at least describe, the changes over time that occur in physical and artificial systems. Approximate controllability of semilinear fractional.
The establishment of a fractional dynamic model is a significant attempt that is expected to reveal the mechanism of fractional quantum system. Click download or read online button to get fractional order control systems book now. We proposed the definition of mittagleffler stability and the fractional lyapunov direct method, which. Using solution operator theory, fractional calculations, fixedpoint technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for nonlocal approximate controllability of semilinear fractional stochastic dynamic systems is formulated and proved by assuming the associated linear system is. The concept of a dynamical system has its origins in newtonian mechanics. Introduction to dynamic systems network mathematics graduate. Nowadays, it is known that the solution to a fractional differential equation cant generally define a dynamical system in the sense of semigroup property due to the history memory induced by the weakly singular kernel. Dynamic analysis of a fractional order r ossler system zhixin cheng. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. Ordinary differential equations and dynamical systems. In recent decades, fractional calculus has found a large number of profound applications, which have triggered the development of both the theory and methods for more reliable discretization and approximations of the dynamics of continuous systems. The theory of fractional differential equations has been extensively studied by many authors 8, 9.
Controllability of nonlinear fractional dynamical systems. Applications of fractional calculus to dynamics of particles, fields and media presents applications of fractional calculus, integral and differential equations of noninteger orders in describing systems with longtime memory, nonlocal spatial and fractal properties. This paper investigates the stability of ndimensional nonlinear fractional differential systems with riemannliouville derivative. The contributions, written by leading researchers in the field, consist of chapters on classical and modern dynamical systems modelled by fractional differential equations in physics, engineering, signal processing, fluid mechanics, and bioengineering. Theory and applications of fractional differential. Complexity evolution of chaotic financial systems based on. The trajectory of a dynamical system can be derived by finding the path for which the fractional poincarecartan 1form. The notions of gradient and hamiltonian systems arise in dynamical systems theory hirsh and smale, 1974. The concept of stab ility is extremely imp ortant because almost every work able control system is design ed to be stable. We discussed fractional nonautonomous systems and the application of the lipschitz condition to fractional order systems. It incorporates the recent upsurge in the general theory of fractional differential equations theory of fractional dynamic systems. School of mathematical science, yancheng teachers university, jiangsu yancheng, 224002, china received july 14, accepted september 8 abstract. The associated geometrical objects have an geometric character. Then we investigate the boundedness and hstability of solutions of caputo fractional differential systems by using fractional comparison principle and fractional lyapunov direct method.
In this paper, we offer a tutorial on fractional calculus in controls. In addition, we have found that chaos exists in the double fractional order chen system. The systems of gradient and hamiltonian type can be considered as a special case of fractional gradient and hamiltonian systems. We introduce the lyapunovkrasovskii approach for fractional neutral systems, which enrich the knowledge of both system theory and fractional calculus.
Stability analysis of distributed order fractional chen system. This paper investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear timevarying caputo fractional dynamic systems of real fractional order possessing internal point delays. Stability of fractionalorder nonlinear dynamic systems. It is based on concepts from mathematics and is a general approach applicable to almost any phenomenon. About robust stability of caputo linear fractional dynamic. The objective of this book is to present a small segment of theory and applications of systems and control governed by ordinary differential equations and inclusions. A simple and elegant scheme of estimating the parameters for such a fractional order process is proposed. Especially, the publications on the theory of fractional differential equations are growing exponentially, but the number of studies on qualitative behaviors of fractional dynamical systems is reduced, and its scope is limited. Apr 23, 2014 fractional dynamic systems are described by fdes, and this special issue consists of 8 original articles covering various aspects of fdes and their applications written by prominent researchers in the field. Fractional dierential equations appear naturally in a number of elds such as physics, engineering, biophysics, blood ow phenomena, aerodynamics, electronanalytical chemistry, biology, control theory, etc. Fractional dynamic systems are described by fdes, and this special issue consists of 8 original articles covering various aspects of fdes and their applications written by prominent researchers in the field. Optimization of fractional order dynamic chemical processing. This problem has been discussed for fractional dynamical systems in 10, 11. Introduction this chapter considers the particle swarm optimization algorithm as a system, whose.
By using the presented inequality, it is shown that the fractional order system is mittagleffler stable if there is a convex and positive definite function such that its fractional order derivative is negative definite. This is a preliminary version of the book ordinary differential equations and dynamical systems. Then, combining the techniques of stochastic analysis theory, fractional calculations and operator semigroup theory, a new set of sufficient conditions for approximate controllability of these equations is. Theory of fractional differential equations in a banach space. Introduction to dynamic systems network mathematics. Mittagleffler stability of fractional order nonlinear. It is expected that any reader who has absorbed the materials presented here would have no difficulty to reach the core of current research. Pdf stability analysis of a class of nonlinear fractional.
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