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

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^𝛼𝑧(𝑡)=𝜌𝑧(𝑡)𝑧(1(0−)=𝑧(𝑧𝑡)_𝑜)=𝜌𝑧(𝑡)−𝜌𝑧²(𝑡),𝑡>0,(1.1)=𝑥_𝑜+𝑖𝑦_𝑜,(1.2) where 𝑧(𝑡)=𝜌𝑥=(𝑎𝑡)++𝑖𝑖𝑏𝑦,(𝑡)𝑎,,𝑏||>𝑧(0𝑡).||≤1,(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^𝛽𝑓(𝑡)=^𝑡0(𝑡−𝑠)^𝛽−1Γ(𝛽)𝑓(𝑠)𝑑𝑠,(1.4) and the Caputo’s definition for the fractional order derivative of order 𝛼∈(0,1] of 𝑓(𝑡) is given by 𝐷^𝛼𝑓(𝑡)=𝐼^1−𝛼𝑑𝑑𝑡𝑓(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 𝐷𝐷^𝛼𝑥(𝑡)=𝑎𝑥(𝑡)−𝑏𝑦(𝑡)−𝑎𝑥²(𝑡)−𝑦²(𝑡)+2𝑏𝑥(𝑡)𝑦(𝑡),𝑡>0,(2.1)_𝛼𝑦(𝑡)=𝑏𝑥(𝑡)+𝑎𝑦(𝑡)−𝑏𝑥²(𝑡)−𝑦²(𝑡)−2𝑎𝑥(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‖=sup_{[]𝑡∈0,𝑇}||𝑥(𝑡)||+sup_{[]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‖^∗=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^𝛼(𝑥,𝑦)^𝜏=+𝑎𝑎𝑥𝑦((𝑡𝑡))−−𝑏𝑏𝑦(𝑥𝑡₂)−𝑎𝑥²(𝑡)−𝑦²(𝑡)+2𝑏𝑥(𝑡)𝑦(𝑡),𝑏𝑥(𝑡)(𝑡)−𝑦²(𝑡)−2𝑎𝑥(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=+(𝑥𝑎(𝑦0()𝑡,)𝑦−(0𝑏))^𝜏𝑥₂+𝐼^𝛼𝑎𝑥(𝑡)−𝑏𝑦(𝑡)−𝑎𝑥²(𝑡)−𝑦²(𝑡)+2𝑏𝑥(𝑡)𝑦(𝑡),𝑏𝑥(𝑡)(𝑡)−𝑦²(𝑡)−2𝑎𝑥(𝑡)𝑦(𝑡)^𝜏.(2.8) Define the operator 𝐹∶𝑋→𝑋 by 𝐹(𝑥(𝑡),𝑦(𝑡))^𝜏+=𝑎𝑦(𝑥(𝑡()0−),𝑏𝑦(𝑥0)₂)^𝜏+𝐼^𝛼𝑎𝑥(𝑡)−𝑏𝑦(𝑡)−𝑎𝑥²(𝑡)−𝑦²(𝑡)+2𝑏𝑥(𝑡)𝑦(𝑡),𝑏𝑥(𝑡)(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) where 𝐾=5(𝑎+𝑏)1𝑁^𝛼.(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]) 𝐼^𝛼𝑎+𝑥𝑎(𝑦𝑡)(𝑡−)−𝑏𝑦𝑏(𝑡)𝑥−₂𝑎𝑥²(𝑡)−𝑦²(𝑡)+2𝑏𝑥(𝑡)𝑦(𝑡),𝑏𝑥(𝑡)(𝑡)−𝑦²(𝑡)−2𝑎𝑥(𝑡)𝑦(𝑡)^𝜏||_𝑡=0=0,(2.12) 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||≤𝛿⇒‖‖(𝑥,𝑦)−𝑥^∗,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,𝑦)−𝑥^∗,𝑦^∗^𝜏‖‖_𝑋≤||𝑥_𝑜−𝑥^∗𝑜||+||𝑦_𝑜−𝑦^∗𝑜||+𝐾‖‖(𝑥,𝑦)−𝑥^∗,𝑦^∗^𝜏‖‖_𝑋,(3.3) which implies that ‖‖(𝑥,𝑦)−(𝑥^∗,𝑦^∗^𝜏‖‖_𝑋≤𝜖=(1(−1−𝐾)𝐾⁻¹)₋₁||𝑥_𝑜−𝑥^∗𝑜||+||𝑦_𝑜−𝑦^∗𝑜||≤𝜖,(3.4)𝛿.(3.5)

4. Equilibrium Points and Their Asymptotic Stability

Let 𝛼∈(0,1] and consider the system ([9, 20–22]) 𝐷𝐷^𝛼_𝛼𝑦₁(𝑡)=𝑓₁𝑦₁,𝑦₂,𝑦₂(𝑡)=𝑓₂𝑦₁,𝑦₂,(4.1) with the initial values 𝑦₁(0)=𝑦_𝑜1,𝑦₂(0)=𝑦_𝑜2.(4.2) To evaluate the equilibrium points, let 𝐷^𝛼𝑦_𝑗(𝑡)=0⇒𝑓_𝑗𝑦^eq₁,𝑦^eq₂=0,𝑗=1,2,(4.3) from which we can get the equilibrium points 𝑦^eq₁,𝑦^eq₂.

To evaluate the asymptotic stability, let 𝑦_𝑗(𝑡)=𝑦^eq_𝑗+𝜀_𝑗(𝑡).(4.4)

So the the equilibrium point (𝑦^eq₁,𝑦^eq₂) is locally asymptotically stable if both the eigenvalues of the Jacobian matrix 𝐴⎡⎢⎢⎢⎣𝜕𝑓¹𝜕𝑦₁𝜕𝑓₁𝜕𝑦₂𝜕𝑓₂𝜕𝑦₁𝜕𝑓₂𝜕𝑦₂⎤⎥⎥⎥⎦(4.5) evaluated at the equilibrium point satisfies (|arg(𝜆₁)|>𝛼𝜋/2,|arg(𝜆₂)|>𝛼𝜋/2) ([9, 20–23]).

For the fractional-order Logistic equation of complex variables consider the following: 𝐷𝐷^𝛼𝑥(𝑡)=𝑎𝑥(𝑡)−𝑏𝑦(𝑡)−𝑎𝑥²(𝑡)−𝑦²(𝑡)+2𝑏𝑥(𝑡)𝑦(𝑡),𝑡>0,_𝛼𝑦(𝑡)=𝑏𝑥(𝑡)+𝑎𝑦(𝑡)−𝑏𝑥²(𝑡)−𝑦²(𝑡)−2𝑎𝑥(𝑡)𝑦(𝑡),𝑡>0.(4.6) To evaluate the equilibrium points, let 𝐷𝐷^𝛼𝑥=0,_𝛼𝑦=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 𝐴=𝑎𝑏−𝑎𝑏(4.8) its eigenvalues are 𝜆=𝑎∓𝑏𝑖.(4.9) A sufficient condition for the local asymptotic stability of the equilibrium point (0,0) is ||arg𝜆₁||>𝛼𝜋2,||arg𝜆²||>𝛼2𝜋,0<𝛼<1,(4.10) that is, 𝑏𝑎>tan𝛼2𝜋(4.11) and 𝑥₀ is small.

For (𝑥_eq,𝑦_eq)=(1,0) we find that 𝐴=−−𝑎𝑏−𝑏𝑎(4.12) 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 𝑥₀ 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 𝑥𝑦((𝑡𝑡))==𝑦𝑥((00))++𝐼𝐼^𝛼_𝛼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),(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>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.

Figure 1 

𝑥(0) = 0.1, 𝑦(0) = 0.9, 𝑎 = 0.1, 𝑏 = 0.9, alpha = 0.8.

Figure 2 

𝑥(0) = 0.1, 𝑦(0) = 0.9, 𝑎 = 0.1, 𝑏 = 0.9, alpha = 0.9.

Figure 3 

𝑥(0) = 0.1, 𝑦(0) = 0.9, 𝑎 = 0.1, 𝑏 = 0.9, alpha = 1.0.

Figure 4 

𝑥(0) = 0.1, 𝑦(0) = 0.9, 𝑎 = 0.1, 𝑏 = 0.9.

Figure 5 

𝑥(0) = 0.2, 𝑦(0) = 0.7, 𝑎 = 0.1, 𝑏 = 0.5, alpha = 0.8.

Figure 6 

𝑥(0) = 0.2, 𝑦(0) = 0.7, 𝑎 = 0.1, 𝑏 = 0.5, alpha = 0.9.

Figure 7 

𝑥(0) = 0.2, 𝑦(0) = 0.7, 𝑎 = 0.1, 𝑏 = 0.5, alpha = 1.0.

Figure 8 

𝑥(0) = 0.2, 𝑦(0) = 0.7, 𝑎 = 0.1, 𝑏 = 0.5.

Figure 9 

𝑥(0) = 0.5, 𝑦(0) = 0.5, 𝑎 = 0.1, 𝑏 = 0.4, alpha = 0.8.

Figure 10 

𝑥(0) = 0.5, 𝑦(0) = 0.5, 𝑎 = 0.1, 𝑏 = 0.4, alpha = 0.9.

Figure 11 

𝑥(0) = 0.5, 𝑦(0) = 0.5, 𝑎 = 0.1, 𝑏 = 0.4, alpha = 1.0.

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.