Chapter 2

DYNAMICS OF FRACTIONAL ORDER

CHAOTIC SYSTEMS

Sachin Bhalekar

Department of Mathematics, Shivaji University, Kolhapur, India

Abstract

This chapter deals with the fractional order generalization of the chaotic system proposed by Li et al [Nonlinear Analysis: Modelling & Control, 18 (2013) 66-77]. We discuss the dynamical properties such as symmetry, dissipativity, stability of equilibrium points and chaos. Further we control the chaos in proposed system and present the synchronization phenomenon.

Keywords: Chaos, fractional order, stability

AMS Subject Classification: 26A33, 65P20, 34L30

1.Introduction

Differential equation is one of the basic tools to model the motion of a physical system. When the solutions of these differential equations are bounded, they either settle down to an equilibrium state or oscillates in a periodic or quasi-periodic state. If the order of the system is three or more and the system is nonlinear then there is a possibility of another type of solutions. Such solutions exhibit aperiodic motion for all time and never settles. These solutions are highly sensitive to initial conditions also. Thus, it is not possible to predict the behavior of the solution for a long time. Such systems are termed as chaotic dynamical systems [1].

If the order of the derivative in the system is not an integer then it is called as fractional order system [2, 3, 4]. There are two very important advantages of the fractional order derivative over a conventional integer order derivative. The first advantage is the nonlocal nature of fractional order derivative. The integer order derivative of a function at a point can be approximated using the nearby values whereas the fractional derivative needs the

E-mail addresses: sbb maths@unishivaji.ac.in, sachin.math@yahoo.co.in

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total history from an initial point. This nonlocality of fractional derivative (FD) is playing a vital role in modelling memory and hereditary properties in the system. Hence the models involving FD are more realistic than those involving integer order ones. Second advantage is the FD is capable of modelling intermediate processes. In some physical problems such as the diffusion of fluid in porous media, the integer order derivative cannot model the real phenomena because the processes are intermediate. In such case, one has to consider a model involving the intermediate (fractional) order derivative.

The generalized dynamical systems involving fractional order derivative are extensively studied by researchers. In the seminal paper, Grigorenko and Grigorenko [5] proposed a fractional order generalization of Lorenz system. Further, various fractional order systems such as Chen system [6], Lu¨ system [7], Rossler system [8]and Liu [9] system are studied in the literature.

In this chapter, we propose a fractional order generalization of the chaotic system proposed by Li et al. [17] and discuss its properties.

2.Preliminaries

2.1.Stability Analysis

We discuss the stability results of the following fractional order system

where 0 < αi < 1 are fractional orders. The system (2.1) is called as a commensurate order if α1 = α2 = · · · = αn otherwise an incommensurate order.

A point p = (x1, x2, · · · , xn) is called an equilibrium point of system (2.1) if fi(p) = 0 for each i = 1, 2, · · · , n.

Theorem 2.1. [10, 11] Consider α = α1 = α2 = · · · = αn in (2.1). An equilibrium point p of the system (2.1) is locally asymptotically stable if all the eigenvalues of the Jacobian matrix

Theorem 2.2. [12, 13] Consider the incommensurate fractional ordered dynamical system given by (2.1). Let αi = vi/ui, (ui, vi) = 1, ui, vi be positive integers. Define M to be the least common multiple of ui's.

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where J is the Jacobian matrix as defined in (2.2) evaluated at point p. Then p is locally asymptotically stable if all the roots of the equation det (Δ(λ)) = 0 satisfy the condition

|arg(λ)| > π/(2M).

Thus an equilibrium point p of the system (2.1) is asymptotically stable if the condition

equilibrium points in fractional order systems (IMFOS). Hence, a necessary condition for fractional order system (2.1) to exhibit chaotic attractor is [12]

2.If D(P ) < 0, a1 ≥ 0, a2 ≥ 0, a3 > 0, α < 2/3 then (2.9) is satisfied.

3.If D(P ) < 0, a1 > 0, a2 > 0, a1a2 = a3 then (2.9) is satisfied for all 0 ≤ α < 1.

2.2.Numerical Method for Solving Fractional Differential Equations

The numerical methods developed for solving equations involving integer order derivatives are not useful to solve those involving fractional derivatives. A modification of Adams- Bashforth-Moulton algorithm is proposed by Diethelm et al. in [14, 15, 16] to solve FDEs.

Consider for α (m − 1, m] the initial value problem (IVP)

The IVP (2.10)–(2.11) is equivalent to the Volterra integral equation

k=0

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26 Sachin Bhalekar

Consider the uniform grid {tn = nh/n = 0, 1, · · · , N} for some integer N and h := T/N. Let yh(tn) be approximation to y(tn). Assume that we have already calculated approximations yh(tj ), j = 1, 2, · · · , n and we want to obtain yh (tn+1) by means of the equation

3.The Model

A new 3D chaotic system proposed by Li et al. in [17] is

where a, b, c and e are constants. We propose a modification in system (3.17) involving fractional order derivatives as below:

where 0 < α ≤ 1. Now we discuss some dynamical properties of the system (3.18).

1.Symmetry. The system is symmetric about z-axis because the transformation (x, y, z) −→ (−x, −y, z) does not change the system.

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The system is dissipative if c < a + b.

3.Non-generalized Lorenz system. The generalized Lorenz system (GLS) discussed in [20] is of the form

where X = [x, y, z]T , λ3 and A is a 2 × 2 matrix with eigenvalues λ1, λ2 such that −λ2 > λ1 > −λ3 > 0. Since our system

is having different structure than the system (3), it is not a GLS.

4. Equilibrium points and their stability. The system (3.18) has three equilibrium

p p

points viz. E0 = (0, 0, 0), E1,2 = −b(a − 2c)/e, −b(a − 2c)/e, −a + 2c . The Jacobian matrix evaluated at an arbitrary point X = (x, y, z) is given by

The characteristic equation of J(E0) is therefore,

P = λ3 + (a + b − c)λ2 + (a2 + ab − 2ac − bc)λ + (a2b − 2abc) = 0. (3.23)

The following expression defines the descriminant of the equation (3.23).

p

Theorem 3.1. (a) If 2c < a < 13/3c then the equilibrium point E0 is asymptotically stable.

(b) If c = 0, b > 0 and a > 0 then E0 is asymptotically stable for α < 2/3.

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