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

and Applied Analysis 3 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∥∥∗ sup t∈ 0,T e−Nt|x t | sup t∈ 0,T e−Nt ∣∣y t ∣∣, N > 0. 2.5 Write the problem 2.1 2.3 in the following matrix form: D ( x, y )τ (ax t − by t − a(x2 t − y2 t ) 2bx t y t , bx t ay t − b ( x2 t − y2 t ) − 2ax t y t )τ , 2.6 and ( x 0 , y 0 )τ (xo, yo)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 Iα(ax t − by t − a(x2 t − y2 t ) 2bx t y t , bx t ay t − b ( x2 t − y2 t ) − 2ax t y t )τ . 2.8 Define the operator F : X → X by F ( x t , y t )τ (x 0 , y 0 )τ Iα(ax t − by t − a(x2 t − y2 t ) 2bx t y t , bx t ay t − b ( x2 t − y2 t ) − 2ax 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, 2.10


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 L ü 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: 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 and the Caputo's definition for the fractional order derivative of order α ∈ 0, 1 of f t is given by

Existence and Uniqueness
The following lemma formulation of the problem can be easily proved.

2.8
Define the operator F : X → X by then by direct calculations, we can get where

2.11
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

Uniform Stability
Theorem 3.1.The solution of the problem 2.6 -2.7 is uniformly stable in the sense that where x * t , y * t is the solution of the differential equation 2.6 with the initial data

4.6
To evaluate the equilibrium points, let D α x 0, D α y 0, 4.7 then x eq , y eq 0, 0 , 1, 0 , are the equilibrium points.For x eq , y eq 0, 0 we find that its eigenvalues are λ a ∓ bi.4.9 A sufficient condition for the local asymptotic stability of the equilibrium point 0, 0 is and x 0 is small.For x eq , y eq 1, 0 we find that 12 its eigenvalues are λ −a ± bi.

4.13
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.

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 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 > tan απ/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 > tan απ/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.

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