Abstract and Applied Analysis

Volume 2012 (2012), Article ID 251715, 12 pages

doi:10.1155/2012/251715

Research Article

Dynamic Properties of the Fractional-Order Logistic Equation of Complex Variables

A. M. A. El-Sayed,¹ E. Ahmed,² and H. A. A. El-Saka³

¹Faculty of Science, Alexandria University, Alexandria 21526, Egypt

²Faculty of Science, Mansoura University, Mansoura 35516, Egypt

³Mathematics Department, Faculty of Science, Damietta University, P.O. Box 34517, New Damietta, Egypt

Received 13 June 2012; Accepted 17 July 2012

Academic Editor: Juan J. Trujillo

Copyright © 2012 A. M. A. El-Sayed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the dynamic properties (equilibrium points, local and global stability, chaos and bifurcation) of the continuous dynamical system of the logistic equation of complex variables. The existence and uniqueness of uniformly Lyapunov stable solution will be proved.

1. Introduction

Dynamical properties and chaos synchronization of deterministic nonlinear systems have been intensively studied over the last two decades on a large number of real dynamical systems of physical nature (i.e., those that involve real variables). However, there are also many interesting cases involving complex variables. As an example, we mention here the complex Lorenz equations, complex Chen and Lü chaotic systems, and some others (see [1–8] and the references therein).

The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers (see [9–16] and references therein).

Consider the following fractional-order Logistic equation of complex variables:

where

Here we study the dynamic properties (equilibrium points, local and global stability, chaos and bifurcation) of the continuous dynamical system of complex variables (1.1)-(1.2). The the existence of a unique uniformly stable solution and the continuous dependence of the solution on the initial data (1.2) are also proved.

Now we give the definition of fractional-order integration and fractional-order differentiation.

Definition 1.1. The fractional integral of order of the function is

and the Caputo’s definition for the fractional order derivative of order of is given by

2. Existence and Uniqueness

The following lemma (formulation of the problem) can be easily proved.

Lemma 2.1. The discontinuous dynamical system (1.1)-(1.2) can be transformed to the system

with the initial values

where and .

Let be the class of continuous functions defined on .

Let be the class of columns vectors with the norm

Let be the class of columns vectors with the equivalent norm

Write the problem (2.1)-(2.3) in the following matrix form:

and

where is the transpose of the matrix.

Now we have the following theorem.

Theorem 2.2. The problem (2.6)-(2.7) has a unique solution .

Proof. Integrating (2.6) -times we obtain

Define the operator by

then by direct calculations, we can get

where

Choose large enough we find that and by the contraction fixed theorem [17] the problem (2.6)-(2.7) has a unique solution .

From the continuity of the solution we deduce that (see [10])

then the solution satisfies the initial condition. Differentiating (2.8), then by the same way as in ([18, 19]), we deduce that the integral equation (2.8) satisfies the problem (2.6)-(2.7) which completes the proof.

3. Uniform Stability

Theorem 3.1. The solution of the problem (2.6)–(2.7) is uniformly stable in the sense that

where is the solution of the differential equation (2.6) with the initial data

Proof. Direct calculations give

which implies that

4. Equilibrium Points and Their Asymptotic Stability

Let and consider the system ([9, 20–22])

with the initial values

To evaluate the equilibrium points, let

from which we can get the equilibrium points .

To evaluate the asymptotic stability, let

So the the equilibrium point is locally asymptotically stable if both the eigenvalues of the Jacobian matrix

evaluated at the equilibrium point satisfies ([9, 20–23]).

For the fractional-order Logistic equation of complex variables consider the following:

To evaluate the equilibrium points, let

then are the equilibrium points.

For we find that

its eigenvalues are

A sufficient condition for the local asymptotic stability of the equilibrium point is

that is,

and is small.

For we find that

its eigenvalues are

A sufficient condition for the local asymptotic stability of the equilibrium point is and is not close to zero.

5. Numerical Methods and Results

An Adams-type predictor-corrector method has been introduced and investigated further in ([24–26]). In this paper we use an Adams-type predictor-corrector method for the numerical solution of fractional integral equation.

The key to the derivation of the method is to replace the original problem (2.1)-(2.2) by an equivalent fractional integral equations

and then apply the PECE (Predict, Evaluate, Correct, Evaluate) method.

The approximate solutions displayed in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 for different . In Figures 1–4 we take and found that the equilibrium point is local asymptotic stable for because the condition is satisfied and the equilibrium point is local asymptotic stable for . In Figures 5–8 we take and found that the equilibrium point is local asymptotic stable for because the condition is satisfied and the equilibrium point is local asymptotic stable for . In Figures 9–12 we take and found that the equilibrium point is local asymptotic stable for .

Figure 1: 0) = 0.1, (0) = 0.9, = 0.1, = 0.9, alpha = 0.8.

Figure 2: (0) = 0.1, (0) = 0.9, = 0.1, = 0.9, alpha = 0.9.

Figure 3: (0) = 0.1, (0) = 0.9, = 0.1, = 0.9, alpha = 1.0.

Figure 4: (0) = 0.1, (0) = 0.9, = 0.1, = 0.9.

Figure 5: (0) = 0.2, (0) = 0.7, = 0.1, = 0.5, alpha = 0.8.

Figure 6: (0) = 0.2, (0) = 0.7, = 0.1, = 0.5, alpha = 0.9.

Figure 7: (0) = 0.2, (0) = 0.7, = 0.1, = 0.5, alpha = 1.0.

Figure 8: (0) = 0.2, (0) = 0.7, = 0.1, = 0.5.

Figure 9: (0) = 0.5, (0) = 0.5, = 0.1, = 0.4, alpha = 0.8.

Figure 10: (0) = 0.5, (0) = 0.5, = 0.1, = 0.4, alpha = 0.9.

Figure 11: (0) = 0.5, (0) = 0.5, = 0.1, = 0.4, alpha = 1.0.

Figure 12: (0) = 0.5, (0) = 0.5, = 0.1, = 0.4.

6. Conclusions

In this paper we considered the fractional-order Logistic equations of complex variables. Here we studied the dynamic properties (equilibrium points, local and global stability, chaos and bifurcation). The existence of a unique uniformly stable solution and the continuous dependence of the solution on the initial data (1.2) are also proved. Also we studied the numerical solution of the system (1.1)-(1.2).

We like to argue that fractional-order equations are more suitable than integer-order ones in modeling biological, economic, and social systems (generally complex adaptive systems) where memory effects are important.

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