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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: 𝐷 𝛼 𝑧 ( 𝑡 ) = 𝜌 𝑧 ( 𝑡 ) ( 1 − 𝑧 ( 𝑡 ) ) = 𝜌 𝑧 ( 𝑡 ) − 𝜌 𝑧 2 ( 𝑡 ) , 𝑡 > 0 , ( 1 . 1 ) 𝑧 ( 0 ) = 𝑧 𝑜 = 𝑥 𝑜 + 𝑖 𝑦 𝑜 , ( 1 . 2 ) where | | | | 𝑧 ( 𝑡 ) = 𝑥 ( 𝑡 ) + 𝑖 𝑦 ( 𝑡 ) , 𝑧 ( 𝑡 ) ≤ 1 , 𝜌 = 𝑎 + 𝑖 𝑏 , 𝑎 , 𝑏 > 0 . ( 1 . 3 ) 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 𝛽 ∈ 𝑅 + of the function 𝑓 ( 𝑡 ) , 𝑡 ∈ 𝐼 is 𝐼 𝛽 𝑓 ( 𝑡 ) = 𝑡 0 ( 𝑡 − 𝑠 ) 𝛽 − 1 Γ ( 𝛽 ) 𝑓 ( 𝑠 ) 𝑑 𝑠 , ( 1 . 4 ) and the Caputo’s definition for the fractional order derivative of order 𝛼 ∈ ( 0 , 1 ] of 𝑓 ( 𝑡 ) is given by 𝐷 𝛼 𝑓 ( 𝑡 ) = 𝐼 1 − 𝛼 𝑑 𝑑 𝑡 𝑓 ( 𝑡 ) . ( 1 . 5 )

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 𝐷 𝛼 𝑥 𝑥 ( 𝑡 ) = 𝑎 𝑥 ( 𝑡 ) − 𝑏 𝑦 ( 𝑡 ) − 𝑎 2 ( 𝑡 ) − 𝑦 2 𝐷 ( 𝑡 ) + 2 𝑏 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) , 𝑡 > 0 , ( 2 . 1 ) 𝛼 𝑦 𝑥 ( 𝑡 ) = 𝑏 𝑥 ( 𝑡 ) + 𝑎 𝑦 ( 𝑡 ) − 𝑏 2 ( 𝑡 ) − 𝑦 2 ( 𝑡 ) − 2 𝑎 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) , 𝑡 > 0 , ( 2 . 2 ) with the initial values 𝑥 ( 0 ) = 𝑥 𝑜 , 𝑦 ( 0 ) = 𝑦 𝑜 , ( 2 . 3 ) where | 𝑥 ( 𝑡 ) | ≤ 1 and | 𝑦 ( 𝑡 ) | ≤ 1 .
Let 𝐶 [ 0 , 𝑇 ] be the class of continuous functions defined on [ 0 , 𝑇 ] .
Let 𝑌 be the class of columns vectors ( 𝑥 ( 𝑡 ) , 𝑦 ( 𝑡 ) ) 𝜏 , 𝑥 , 𝑦 ∈ 𝐶 [ 0 , 𝑇 ] with the norm ‖ ( 𝑥 , 𝑦 ) 𝜏 ‖ 𝑌 = ‖ 𝑥 ‖ + ‖ 𝑦 ‖ = s u p [ ] 𝑡 ∈ 0 , 𝑇 | | | | 𝑥 ( 𝑡 ) + s u p [ ] 𝑡 ∈ 0 , 𝑇 | | | | . 𝑦 ( 𝑡 ) ( 2 . 4 ) Let 𝑋 be the class of columns vectors ( 𝑥 ( 𝑡 ) , 𝑦 ( 𝑡 ) ) 𝜏 , 𝑥 , 𝑦 ∈ 𝐶 [ 0 , 𝑇 ] with the equivalent norm ‖ ( 𝑥 , 𝑦 ) 𝜏 ‖ 𝑋 = ‖ 𝑥 ‖ ∗ + ‖ 𝑦 ‖ ∗ = s u p [ ] 𝑡 ∈ 0 , 𝑇 𝑒 − 𝑁 𝑡 | | | | 𝑥 ( 𝑡 ) + s u p [ ] 𝑡 ∈ 0 , 𝑇 𝑒 − 𝑁 𝑡 | | | | 𝑦 ( 𝑡 ) , 𝑁 > 0 . ( 2 . 5 ) Write the problem (2.1)-(2.3) in the following matrix form: 𝐷 𝛼 ( 𝑥 , 𝑦 ) 𝜏 = 𝑥 𝑎 𝑥 ( 𝑡 ) − 𝑏 𝑦 ( 𝑡 ) − 𝑎 2 ( 𝑡 ) − 𝑦 2 𝑥 ( 𝑡 ) + 2 𝑏 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) , 𝑏 𝑥 ( 𝑡 ) + 𝑎 𝑦 ( 𝑡 ) − 𝑏 2 ( 𝑡 ) − 𝑦 2 ( 𝑡 ) − 2 𝑎 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) 𝜏 , ( 2 . 6 ) and ( 𝑥 ( 0 ) , 𝑦 ( 0 ) ) 𝜏 = 𝑥 𝑜 , 𝑦 𝑜 𝜏 , ( 2 . 7 ) where 𝜏 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 ( 𝑥 , 𝑦 ) ∈ 𝑋 .

Proof. Integrating (2.6) 𝛼 -times we obtain ( 𝑥 ( 𝑡 ) , 𝑦 ( 𝑡 ) ) 𝜏 = ( 𝑥 ( 0 ) , 𝑦 ( 0 ) ) 𝜏 + 𝐼 𝛼 𝑥 𝑎 𝑥 ( 𝑡 ) − 𝑏 𝑦 ( 𝑡 ) − 𝑎 2 ( 𝑡 ) − 𝑦 2 𝑥 ( 𝑡 ) + 2 𝑏 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) , 𝑏 𝑥 ( 𝑡 ) + 𝑎 𝑦 ( 𝑡 ) − 𝑏 2 ( 𝑡 ) − 𝑦 2 ( 𝑡 ) − 2 𝑎 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) 𝜏 . ( 2 . 8 ) Define the operator 𝐹 ∶ 𝑋 → 𝑋 by 𝐹 ( 𝑥 ( 𝑡 ) , 𝑦 ( 𝑡 ) ) 𝜏 = ( 𝑥 ( 0 ) , 𝑦 ( 0 ) ) 𝜏 + 𝐼 𝛼 𝑥 𝑎 𝑥 ( 𝑡 ) − 𝑏 𝑦 ( 𝑡 ) − 𝑎 2 ( 𝑡 ) − 𝑦 2 𝑥 ( 𝑡 ) + 2 𝑏 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) , 𝑏 𝑥 ( 𝑡 ) + 𝑎 𝑦 ( 𝑡 ) − 𝑏 2 ( 𝑡 ) − 𝑦 2 ( 𝑡 ) − 2 𝑎 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) 𝜏 , ( 2 . 9 ) then by direct calculations, we can get ‖ 𝐹 ( 𝑥 , 𝑦 ) − 𝐹 ( 𝑢 , 𝑣 ) 𝜏 ‖ 𝑋 ≤ 𝐾 ‖ ( 𝑥 , 𝑦 ) − ( 𝑢 , 𝑣 ) 𝜏 ‖ 𝑋 , ( 2 . 1 0 ) where 1 𝐾 = 5 ( 𝑎 + 𝑏 ) 𝑁 𝛼 . ( 2 . 1 1 ) Choose 𝑁 large enough we find that 𝐾 < 1 and by the contraction fixed theorem [17] the problem (2.6)-(2.7) has a unique solution ( 𝑥 , 𝑦 ) ∈ 𝑋 .
From the continuity of the solution we deduce that (see [10]) 𝐼 𝛼 𝑥 𝑎 𝑥 ( 𝑡 ) − 𝑏 𝑦 ( 𝑡 ) − 𝑎 2 ( 𝑡 ) − 𝑦 2 𝑥 ( 𝑡 ) + 2 𝑏 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) , 𝑏 𝑥 ( 𝑡 ) + 𝑎 𝑦 ( 𝑡 ) − 𝑏 2 ( 𝑡 ) − 𝑦 2 ( 𝑡 ) − 2 𝑎 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) 𝜏 | | 𝑡 = 0 = 0 , ( 2 . 1 2 ) 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 | | 𝑥 𝑜 − 𝑥 ∗ 𝑜 | | + | | 𝑦 𝑜 − 𝑦 ∗ 𝑜 | | ‖ ‖ 𝑥 ≤ 𝛿 ⇒ ( 𝑥 , 𝑦 ) − ∗ , 𝑦 ∗ ‖ ‖ 𝑋 ≤ 𝜖 , ( 3 . 1 ) where ( 𝑥 ∗ ( 𝑡 ) , 𝑦 ∗ ( 𝑡 ) ) is the solution of the differential equation (2.6) with the initial data ( 𝑥 ( 0 ) , 𝑦 ( 0 ) ) 𝜏 = 𝑥 ∗ 𝑜 , 𝑦 ∗ 𝑜 𝜏 . ( 3 . 2 )

Proof. Direct calculations give ‖ ‖ 𝑥 ( 𝑥 , 𝑦 ) − ∗ , 𝑦 ∗ 𝜏 ‖ ‖ 𝑋 ≤ | | 𝑥 𝑜 − 𝑥 ∗ 𝑜 | | + | | 𝑦 𝑜 − 𝑦 ∗ 𝑜 | | ‖ ‖ 𝑥 + 𝐾 ( 𝑥 , 𝑦 ) − ∗ , 𝑦 ∗ 𝜏 ‖ ‖ 𝑋 , ( 3 . 3 ) which implies that ‖ ‖ 𝑥 ( 𝑥 , 𝑦 ) − ( ∗ , 𝑦 ∗ 𝜏 ‖ ‖ 𝑋 ≤ ( 1 − 𝐾 ) − 1 | | 𝑥 𝑜 − 𝑥 ∗ 𝑜 | | + | | 𝑦 𝑜 − 𝑦 ∗ 𝑜 | | ≤ 𝜖 , ( 3 . 4 ) 𝜖 = ( 1 − 𝐾 ) − 1 𝛿 . ( 3 . 5 )

4. Equilibrium Points and Their Asymptotic Stability

Let 𝛼 ∈ ( 0 , 1 ] and consider the system ([9, 20–22]) 𝐷 𝛼 𝑦 1 ( 𝑡 ) = 𝑓 1 𝑦 1 , 𝑦 2 , 𝐷 𝛼 𝑦 2 ( 𝑡 ) = 𝑓 2 𝑦 1 , 𝑦 2 , ( 4 . 1 ) with the initial values 𝑦 1 ( 0 ) = 𝑦 𝑜 1 , 𝑦 2 ( 0 ) = 𝑦 𝑜 2 . ( 4 . 2 ) To evaluate the equilibrium points, let 𝐷 𝛼 𝑦 𝑗 ( 𝑡 ) = 0 ⇒ 𝑓 𝑗 𝑦 e q 1 , 𝑦 e q 2 = 0 , 𝑗 = 1 , 2 , ( 4 . 3 ) from which we can get the equilibrium points 𝑦 e q 1 , 𝑦 e q 2 .

To evaluate the asymptotic stability, let 𝑦 𝑗 ( 𝑡 ) = 𝑦 e q 𝑗 + 𝜀 𝑗 ( 𝑡 ) . ( 4 . 4 )

So the the equilibrium point ( 𝑦 e q 1 , 𝑦 e q 2 ) is locally asymptotically stable if both the eigenvalues of the Jacobian matrix 𝐴 ⎡ ⎢ ⎢ ⎢ ⎣ 𝜕 𝑓 1 𝜕 𝑦 1 𝜕 𝑓 1 𝜕 𝑦 2 𝜕 𝑓 2 𝜕 𝑦 1 𝜕 𝑓 2 𝜕 𝑦 2 ⎤ ⎥ ⎥ ⎥ ⎦ ( 4 . 5 ) evaluated at the equilibrium point satisfies ( | a r g ( 𝜆 1 ) | > 𝛼 𝜋 / 2 , | a r g ( 𝜆 2 ) | > 𝛼 𝜋 / 2 ) ([9, 20–23]).

For the fractional-order Logistic equation of complex variables consider the following: 𝐷 𝛼 𝑥 𝑥 ( 𝑡 ) = 𝑎 𝑥 ( 𝑡 ) − 𝑏 𝑦 ( 𝑡 ) − 𝑎 2 ( 𝑡 ) − 𝑦 2 𝐷 ( 𝑡 ) + 2 𝑏 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) , 𝑡 > 0 , 𝛼 𝑦 𝑥 ( 𝑡 ) = 𝑏 𝑥 ( 𝑡 ) + 𝑎 𝑦 ( 𝑡 ) − 𝑏 2 ( 𝑡 ) − 𝑦 2 ( 𝑡 ) − 2 𝑎 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) , 𝑡 > 0 . ( 4 . 6 ) To evaluate the equilibrium points, let 𝐷 𝛼 𝐷 𝑥 = 0 , 𝛼 𝑦 = 0 , ( 4 . 7 ) then ( 𝑥 e q , 𝑦 e q ) = ( 0 , 0 ) , ( 1 , 0 ) , are the equilibrium points.

For ( 𝑥 e q , 𝑦 e q ) = ( 0 , 0 ) we find that 𝐴 = 𝑎 − 𝑏 𝑏 𝑎 ( 4 . 8 ) its eigenvalues are 𝜆 = 𝑎 ∓ 𝑏 𝑖 . ( 4 . 9 ) A sufficient condition for the local asymptotic stability of the equilibrium point ( 0 , 0 ) is | | 𝜆 a r g 1 | | > 𝛼 𝜋 2 , | | 𝜆 a r g 2 | | > 𝛼 𝜋 2 , 0 < 𝛼 < 1 , ( 4 . 1 0 ) that is, 𝑏 𝑎 > t a n 𝛼 𝜋 2 ( 4 . 1 1 ) and 𝑥 0 is small.

For ( 𝑥 e q , 𝑦 e q ) = ( 1 , 0 ) we find that 𝐴 = − 𝑎 𝑏 − 𝑏 − 𝑎 ( 4 . 1 2 ) its eigenvalues are 𝜆 = − 𝑎 ± 𝑏 𝑖 . ( 4 . 1 3 ) A sufficient condition for the local asymptotic stability of the equilibrium point ( 1 , 0 ) is 𝑎 > 0 and 𝑥 0 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 𝑥 ( 𝑡 ) = 𝑥 ( 0 ) + 𝐼 𝛼 𝑥 𝑎 𝑥 ( 𝑡 ) − 𝑏 𝑦 ( 𝑡 ) − 𝑎 2 ( 𝑡 ) − 𝑦 2 , 𝑦 ( 𝑡 ) + 2 𝑏 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) ( 𝑡 ) = 𝑦 ( 0 ) + 𝐼 𝛼 𝑥 𝑏 𝑥 ( 𝑡 ) + 𝑎 𝑦 ( 𝑡 ) − 𝑏 2 ( 𝑡 ) − 𝑦 2 , ( 𝑡 ) − 2 𝑎 𝑥 ( 𝑡 ) 𝑦 ( 𝑡 ) ( 5 . 1 ) 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 0 < 𝛼 ≤ 1 . In Figures 1–4 we take 𝑥 ( 0 ) = 0 . 1 , 𝑦 ( 0 ) = 0 . 9 , 𝑎 = 0 . 1 , 𝑏 = 0 . 9 and found that the equilibrium point ( 0 , 0 ) is local asymptotic stable for 𝛼 = 0 . 8 , 0 . 9 because the condition 𝑏 / 𝑎 > t a n ( 𝛼 𝜋 / 2 ) is satisfied and the equilibrium point ( 1 , 0 ) is local asymptotic stable for 𝛼 = 1 . 0 . In Figures 5–8 we take 𝑥 ( 0 ) = 0 . 2 , 𝑦 ( 0 ) = 0 . 7 , 𝑎 = 0 . 1 , 𝑏 = 0 . 5 and found that the equilibrium point ( 0 , 0 ) is local asymptotic stable for 𝛼 = 0 . 8 because the condition 𝑏 / 𝑎 > t a n ( 𝛼 𝜋 / 2 ) is satisfied and the equilibrium point ( 1 , 0 ) is local asymptotic stable for 𝛼 = 0 . 9 , 1 . 0 . In Figures 9–12 we take 𝑥 ( 0 ) = 0 . 5 , 𝑦 ( 0 ) = 0 . 5 , 𝑎 = 0 . 1 , 𝑏 = 0 . 4 and found that the equilibrium point ( 1 , 0 ) is local asymptotic stable for 𝛼 = 0 . 8 , 0 . 9 , 1 . 0 .

251715.fig.001
Figure 1: 𝑥 ( 0) = 0.1, 𝑦 (0) = 0.9, 𝑎 = 0.1, 𝑏 = 0.9, alpha = 0.8.
251715.fig.002
Figure 2: 𝑥 (0) = 0.1, 𝑦 (0) = 0.9, 𝑎 = 0.1, 𝑏 = 0.9, alpha = 0.9.
251715.fig.003
Figure 3: 𝑥 (0) = 0.1, 𝑦 (0) = 0.9, 𝑎 = 0.1, 𝑏 = 0.9, alpha = 1.0.
251715.fig.004
Figure 4: 𝑥 (0) = 0.1, 𝑦 (0) = 0.9, 𝑎 = 0.1, 𝑏 = 0.9.
251715.fig.005
Figure 5: 𝑥 (0) = 0.2, 𝑦 (0) = 0.7, 𝑎 = 0.1, 𝑏 = 0.5, alpha = 0.8.
251715.fig.006
Figure 6: 𝑥 (0) = 0.2, 𝑦 (0) = 0.7, 𝑎 = 0.1, 𝑏 = 0.5, alpha = 0.9.
251715.fig.007
Figure 7: 𝑥 (0) = 0.2, 𝑦 (0) = 0.7, 𝑎 = 0.1, 𝑏 = 0.5, alpha = 1.0.
251715.fig.008
Figure 8: 𝑥 (0) = 0.2, 𝑦 (0) = 0.7, 𝑎 = 0.1, 𝑏 = 0.5.
251715.fig.009
Figure 9: 𝑥 (0) = 0.5, 𝑦 (0) = 0.5, 𝑎 = 0.1, 𝑏 = 0.4, alpha = 0.8.
251715.fig.0010
Figure 10: 𝑥 (0) = 0.5, 𝑦 (0) = 0.5, 𝑎 = 0.1, 𝑏 = 0.4, alpha = 0.9.
251715.fig.0011
Figure 11: 𝑥 (0) = 0.5, 𝑦 (0) = 0.5, 𝑎 = 0.1, 𝑏 = 0.4, alpha = 1.0.
251715.fig.0012
Figure 12: 𝑥 (0) = 0.5, 𝑦 (0) = 0.5, 𝑎 = 0.1, 𝑏 = 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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