Dynamics and Stability of Some Classes of Fractional Order Systems

Lenka, Bichitra Kumar (2019) Dynamics and Stability of Some Classes of Fractional Order Systems. PhD thesis, Indian Institute Of Science Education And Research Kolkata.

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Abstract

Calculus has served a major role in different areas of mathematics, physics, engineering and in many other areas of science. However in the last few decades, the concept of fractional calculus has emerged as an important branch of mathematics which deals with the generalization of the concepts of integer order calculus to the non-integer order calculus. Fractional calculus has been applied in various areas of science and technology. It has been found that fractional derivative or integral operators have the capability to model various applied problems that can be described by a set of fractional order systems. However, finding the solutions to fractional order systems is generally difficult. Thus, it is often necessary to solve and analyse the local and global behaviours of such fractional order systems. In this thesis, we investigate some qualitative and quantitative properties of the initial value problems of fractional order systems in the sense of Caputo and Riemann-Liouville fractional derivative. At first, we consider the initial value problem of linear fractional order systems. By using the Laplace transform method, we derive analytic explicit solutions to systems where its state equations contain arbitrary order fractional orders. Based on the concept matrix diagonalization and partial fractional decomposition, we discuss the structures and give the explicit formulas for the solutions to such fractional order systems and express these in terms of Mittag-Leffler functions for both Caputo and Riemann-Liouville fractional derivative cases. However, it is often difficult to find the asymptotic behaviours of the solutions to such systems by knowing the explicit solutions. We employ the Laplace final value theorem to investigate the asymptotic stability of autonomous linear fractional system for both Caputo and Riemann-Liouville fractional derivative cases whenever the fractional orders in its state equations lie between 0 and 2. When it comes to the nonlinear fractional order systems, the investigation of stability becomes much more difficult. Thus, we consider the initial value problem of nonlinear fractional order systems to both Caputo and Riemann-Liouville fractional order cases whenever the fractional orders in its state equations lie between 0 and 2. Based on the asymptotic behaviours of the stability of linear fractional order systems, at first, we propose linearization theorems for autonomus nonlinear fractional order systems to both Riemann-Liouville and Caputo fractional derivative cases. Then, we consider a class of nonautonomous nonlinear fractional order system. By utilizing the Laplace transform, generalized Gronwall-inequality and the bounds of Mittag-Leffler function, we present stability conditions of such a class of systems. We discuss the stability of the equilibrium points of such systems by presenting a few illustrative examples. Do such systems exhibit interesting dynamical behaviours? Motivated by this question, we consider the fractional order generalization of the initial value problem of famous Lorenz system where the fractional orders in its state equations lie between 0 and 2. By varying the fractional orders in its state equations, we have found that such a system exhibits stable, oscillatory, and chaotic behaviours as well as coexistence of such behaviours for different values of fractional orders as well as for the different choices of initial conditions. Due to the appearance of the uncertain behaviours of such fractional order systems, it is often necessary to control the behaviours of such fractional order systems. By utilizing the linear state feedback control scheme, we design the control strategy for stabilizing the chaotic behaviours to the unstable points such systems. The discussions and the results presented in our work can be utilized for understanding the dynamics as well as for the analysis of fractional order systems. Thus, the thesis work opens up new opportunities and challenges for the further investigation of the stability of nonautonomous fractional order systems in many ways.

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