Dynamic behavior of fractional order Duffing chaotic system and its ...

Appl. Math. Mech. -Engl. Ed., 33(5), 567–582 (2012) DOI 10.1007/s10483-012-1571-6 c Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Applied Mathematics and Mechanics (English Edition)

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control∗ Gui-tian HE (), Mao-kang LUO () (Department of Mathematics, Sichuan University, Chengdu 610064, P. R. China)

Abstract With the increasingly deep studies in physics and technology, the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research. In this paper, the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively investigated. With the stability criterion of linear fractional systems, the synchronization of a fractional non-autonomous system is obtained. Specifically, an effective singly active control is proposed and used to synchronize a fractional order Duffing system. The numerical results demonstrate the effectiveness of the proposed methods. Key words nization

Caputo fractional derivative, fractional order Duffing system, synchro-

Chinese Library Classification O415.5, O151, O369, O193 2010 Mathematics Subject Classification 34A34, 93C83

1

Introduction

The history of fractional calculus is more than 300 years, but only in recent decades did the applied scientists and the engineers realize that such fractional differential equations provided a good approach to describe the complex phenomena in nature, such as non-Brownian motion, signal processing, system identification, control, viscoelastic material, and polymer. Compared with the classical integer order models, the fractional order derivative provides a better instrument for the description of memory and hereditary properties of various materials and processes[1–3] . The study of chaos in fractional order dynamical systems and related systems, such as fractionally damped Duffing systems[4] , fractional order modified Duffing systems[5] , and fractional order Lorenz systems[6] , is receiving growing attention[4–9] . Chaos synchronization, another important topic in nonlinear science, has extensive applications in vast areas of physics and engineering science[10–16] . Therefore, the synchronization of fractional order chaotic systems starts to attract increasing attention and becomes an interesting problem in secure communication and control processing[17–21] . Duffing systems have a wide range of applications. One of the most important aspects is weak signal detection, which is due to their sensitivity to weak signals[22–24] . In our work, the ∗ Received Oct. 30, 2011 / Revised Feb. 5, 2012 Project supported by the National Natural Science Foundation of China (No. 11171238) and the Program for Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (No. IRTO0742) Corresponding author Mao-kang LUO, Professor, Ph. D., E-mail: [email protected]

568

Gui-tian HE and Mao-kang LUO

dynamic properties of a fractional order Duffing system is studied. The existence and uniqueness about the fractional order Duffing system are established, and the stability of equilibria of the system is analyzed. This paper is organized as follows. In Section 2, the definition of the fractional calculus, some basic properties for fractional order systems, and the numerical algorithm for fractional order systems are presented. In Section 3, the fractional order Duffing system is introduced. In Section 4, the existence of the solutions and the stability of the equilibria for the fractional Duffing system are analyzed. In Section 5, the synchronization of non-autonomous systems is investigated. The active control technique is applied to synchronize the fractional order nonautonomous systems. Especially, the single active control technique is applied to synchronize the fractional order Duffing system. Finally, conclusions are drawn in Section 6.

2

Preliminaries

2.1 Fractional calculus There are several definitions for fractional derivatives/integrals[1–3] . The Caputo definition and the Riemann-Liouville definition are two well-known definitions. Definition 1 A real function f (t) (t > 0) is said to be in the space Cα (α ∈ R) if there exists a real number p (> α) such that f (t) = tp f1 (t),

f1 (t) ∈ C[0, ∞].

Definition 2 A real function f (t) (t > 0) is said to be in the space Cαm (m ∈ N ∪ {0}) if f (m) ∈ Cα . Definition 3 Let f ∈ Cα and α −1. Then, the Riemann-Liouville integral of the order μ (> 0) is given by t 1 I μ f (t) = (t − τ )μ−1 f (τ )dτ, t > 0, (1) Γ(μ) 0 where Γ(·) is the Gamma function. Definition 4 The Caputo fractional derivative of f (f ∈ C1m and m ∈ N ∪ {0}) is defined by D∗q f (t) = I m−q f (m) (t),

q > 0,

(2)

where m = q, i.e., m is the largest integer which is no less than q. The Caputo fractional operator D∗q is commutable, i.e., D∗p D∗q f (x) = D∗q D∗p f (x) = D∗p+q f (x),

∀p, q ∈ R+

if f (x) is sufficiently smooth. Definition 5 The Laplace transform of the Caputo fractional derivative is L{D∗q f (t)}

q

= s L{f (t)} −

m−1

sq−1−k f (k) (0),

(3)

k=0

where L means the Laplace transform, and s is a complex variable. Upon considering the initial conditions to zero, Eq. (3) reduces to L{D∗q f (t)} = sq L{f (t)}.

(4)

Dynamic behavior of fractional order Duffing chaotic system and its synchronization

2.2 Existence of solution and stability theorems The initial value problem of the fractional order equation is q D∗ x(t) = f (t, x(t)),

(5)

x(0) = x0 . Proposition 1[1, 25]

569

Assume that D : [0 : T ∗ ] × [x0 − δ, x0 + δ]

with some T ∗ > 0 and some δ > 0. Let f : D :→ R be bounded on D and fulfill a Lipschitz condition with respect to the second variable, i.e., |f (t, x) − f (t, y)| L|x − y| with some constant number L > 0 independent of t, x, and y. Then, there exists at most one function x : [0, T ] → R solving the initial value problem, where δΓ(q + 1) q1 . T := min T ∗ , f Definition 6[26] Proposition 2

The system (5) is said to be asymptotically stable if lim x(t) = 0.

[27–28]

t→∞

For a given linear autonomous fractional order system D∗q X(t) = AX(t),

X(0) = X0

(6)

with X(t) = (x1 , x2 , · · · , xn )T ∈ Rn ,

A = (aij ) ∈ Rn×n ,

q = (q1 , q2 , · · · , qn )T ,

where 0 < qi 2 (i = 1, 2, · · · , n), assume m to be the lowest common multiple of the denominators ui of qi satisfying qi = Set γ =

1 m.

vi , ui

(ui , vi ) = 1,

ui , vi ∈ Z+ .

Define the following characteristic equation: det(diag(λmq1 , λmq2 , · · · , λmqn ) − A) = 0.

Then, the zero solution to the system (6) is globally asymptotically stable if all roots λi of the characteristic equation satisfy π |arg(λi )| > γ. 2 Definition 7 The fractional order equation (5) is a commensurate fractional order system if the condition q1 = q2 = · · · = qn = q is satisfied. The fractional order equation (5) is an incommensurate fractional order system if the condition q1 = q2 = · · · = qn is satisfied. Proposition 3[26, 28] For a given incommensurate fractional order nonlinear system qi D∗ xi (t) = fi (x1 (t), x2 (t), · · · , xn (t)), (7) xi (0) = ci , i = 1, 2, · · · , n,

570


where 0 < qi 2 (i = 1, 2, · · · , n), suppose that m is the lowest common multiple of the denominators ui of qi satisfying qi =

vi , ui

(ui , vi ) = 1,

ui , vi ∈ Z+ .

1 . Then, the system (7) is asymptotically stable if |arg(λ)| > Set γ = m following equation: det(diag(λmq1 λmq2 · · · λmqn ) − J) = 0,

where J=

π 2γ

for all roots λ in the

∂f . ∂x

2.3 Numerical algorithm for solving fractional order systems The numerical algorithm for solving the fractional order differential equation is proposed by Diethelm and Ford[29] and Diethelm et al.[30] . This scheme is the generalization of the AdamsBashforth-Moulton method, i.e., the prediector-corrector approach. The initial value problem of the fractional order equation is

D∗q y(t) = f (t, y(t)), (k)

y (k) (0) = y0 ,

0 t T, (8)

k = 0, 1, · · · , m − 1.

It is equivalent to the Voltergral integral equation q−1

y(t) =

(k) t y0

1 + k! Γ(α)

k=0

Set h=

k

T , N

tn = nh,

0

t

(t − s)q−1 f (s, y(s))ds.

(9)

n = 0, 1, 2, · · · , N.

Then, Eq. (9) can be discreted as follows: q−1

yh (tn+1 ) =

(k) t

y0

k

k!

k=0

+

hq f (tn+1 , yhp (tn+1 )) Γ(q + 2)

n

+

hq aj,n+1 f (tj , yh (tj )), Γ(q + 2) j=0

(10)

where

αj,n+1

⎧ q+1 n − (n − q)(n + 1)q , j = 0, ⎪ ⎪ ⎨ = (n − j + 2)q+1 + (n − j)q+1 − 2(n − j + 1)q+1 , ⎪ ⎪ ⎩ 1, j = n + 1.

1 j n,

(11)

The preliminary approximation yhp (tn+1 ) is called the predictor and is given by yhp (tn+1 )

q−1

=

k=0

(k) t

y0

n

k

k!

+

1 bj,n+1 f (tj , yh (tj )), Γ(q) j=0

(12)


571

where βj,n+1 =

hq ((n + 1) − j)q − (n − j)q . q

(13)

|y(tj ) − yh (tj )| = O(hp ),

(14)

The error in this method is max

j=0,1,··· ,N

where p = min(2, 1 + q).

3

Fractional order Duffing systems The famous Duffing system[22] is x ¨ + k x˙ − ax + bx3 = r cos(ωt),

(15)

where k, a, b, r, and ω are positive constant parameters. It can be written as two first-order differential equations x˙ = y, (16)

y˙ = −ky + ax − bx3 + r cos(ωt). Now, we consider the fractional order Duffing system D∗α x + kD∗β x + ax − bx3 = r cos(ωt),

1 < α 2,

0 < β < α.

It can be written as two fractional order differential equations β D∗ x = y, D∗α−β y = −ky + ax − bx3 + r cos(ωt). Let q1 = β and q2 = α − β. Then, Eq. (18) is equivalent to q1 D∗ x = y, 0 < q1 < 2, D∗q2 y = −ky + ax − bx3 + r cos(ωt),

4

0 < q2 < 2.

(17)

(18)

(19)

Existence of solution and its stability

Theorem 1 The initial value problem of the fractional order Duffing system can be represented in the following form: q D∗ X(t) = AX(t) + F (t) + x21 (t)BX(t), t ∈ [0, M ], (20) X(0) = X0 , where some constant M > 0, and ⎧ X(t) = (x1 (t), x2 (t))T ∈ R2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X0 = (x10 , x20 )T , q = (q1 , q2 )T , ⎪ ⎪ ⎨ 0 0 0 1 , , B = A = ⎪ −b 0 a −k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎩ F (t) = . r cos(ωt)

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Then, the system (20) has a unique solution. Proof Let G(t, X(t)) = AX(t) + x21 (t)BX(t) + F (t). It is obviously continuous and bounded on the interval [X0 − δ, X0 + δ] for any δ > 0. Furthermore, one has |G(t, X(t)) − G(t, Y (t))| |A(X(t) − Y (t))| + |x21 (t)BX(t) − y12 (t)BY (t)| since |x21 (t)BX(t)

−

y12 (t)BY

0

(t)| = b(y1 − x1 )(y12 + x1 y1 + x21 ) b|y12 + x1 y1 + x21 ||x1 − y1 | 3b(|X0 | + δ)2 |x1 − y1 | 3b(|X0 | + δ)2 |X(t) − Y (t)|,

|G(t, X(t)) − G(t, Y (t))| L|X(t) − Y (t)|, where | · | and · denote the vector norm and the matrix norm, respectively, and L = A + 3b(|X0 | + δ)2 ,

X(t), Y (t) ∈ R2 .

The above inequality manifests that G(t, X(t)) satisfies the Lipschitz condition. Based on Proposition 1, we can conclude that the initial value problem of the fractional order Duffing system has a unique solution. It is easy to check that Eq. (19) possesses three fixed points, i.e., O(0, 0) and P± (± ab , 0). Theorem 2 With respect to the system (19), we have (i) the equilibrium O is unstable; (ii) the equilibrium P+ and P− are all asymptotically stable if ⎧ k2 ⎪ ⎪ a > , ⎨ k2 8 or a √ ⎪ 2 8 ⎪ ⎩ 8a − k > tan π γ k 2 is satisfied.

0 1 is the Jacobian matrix of the system (19) at the equilibrium a −k O, which corresponding eigenvalues are √ √ −k + k 2 + 4a −k − k 2 + 4a λ1 = , λ2 = . 2 2 Proof

(i) J0 =

We know that λ1 > 0. Then, |arg λ1 | = 0

π γ. 2

Therefore, the system (19) is asymptotically stable. 2 If a > k8 , the corresponding eigenvalues are λ1,2 =

√ −k ± i 8a − k 2 . 2

Then, P± are asymptotically stable if √ π 8a − k 2 > tan γ . k 2 Remark 1

The equilibrium P+ and P− are unstable if tan

π √8a − k 2 γ . 2 k

Therefore, a necessary condition for the system (19) to exhibit a bifurcation or chaotic attractor is that it satisfies the above inequality.

5

Synchronization of fractional order non-autonomous systems

Consider the master-slave (or drive-response) synchronization scheme of two non-autonomous different fractional order systems: q D∗ X(t) = f (t, X(t)), (21) D∗q Y (t) = g(t, Y (t)) + U (t), where q = (q1 , q2 , · · · , qn )T , and X ∈ Rn and Y ∈ Rn represent the states of the drive and response systems, respectively. f : Rn → Rn and g : Rn → Rn are the vector fields of the drive and response systems, respectively. The aim is to choose a suitable line control function Ut = (u1 , u2 , · · · , un ) such that the states of the drive and response systems are synchronized, i.e., lim Y − X = 0, where · is the Euclidean norm. t→∞

5.1 Synchronization of fractional order Duffing system In this subsection, the goal is to achieve the chaos synchronization of the fractional order (including incommensurate order and commensurate order) Duffing system by using active control. The drive and response systems are given as follows: q1 D∗ x1 = y1 , (22) D∗q2 y1 = −ky1 + ax1 − bx31 + r cos(ωt), q1 D∗ x2 = y2 + u1 , (23) D∗q2 y2 = −ky2 + ax2 − bx32 + r cos(ωt) + u2 .

574


The unknown terms u1 and u2 are active control functions to be determined. Define the error functions as e1 = x2 − x1 , (24) e2 = y2 − y1 . By subtracting Eq. (22) from Eq. (23) and using Eq. (24), we obtain q1 D∗ e1 = e2 + u 1 , D∗q2 e2 = −ke2 + ae1 − b(x32 − x31 ) + u2 . Let

u1 = −e2 − e1 ,

(25)

(26)

u2 = −ae1 + b(x32 − x31 ). The error dynamic systems can be rewritten as q1 D∗ e1 = −e1 ,

(27)

D∗q2 e2 = −ke2 .

Theorem 3 For any initial conditions, the two systems (Eqs. (22) and (23)) are globally asymptotically synchronized with the control law (28). Proof Assume that qi ∈ (1, 2) (i = 1, 2). For the other case, the proof is similar to the former. By taking the Laplace transform in both sides of Eq. (25), letting Ei (s) = L{(ei )} (i = 1, 2) and applying Eq. (3), we obtain ⎧ ⎨ sq1 E1 (s) − sq1 −1 e1 (0) − sq1 −2 e(1) 1 (0) = −E1 (s), (28) ⎩ sq2 E (s) − sq2 −1 e (0) − sq2 −2 e(1) (0) = −kE (s). 2 2 2 2 From Eq. (28), we can easily obtain ⎧ (1) ⎪ sq1 −1 e1 (0) + sq1 −2 e1 (0) ⎪ ⎪ , ⎨ E1 (s) = q s 1 +1

(29)

(1) ⎪ q2 −1 ⎪ e2 (0) + sq2 −2 e2 (0) ⎪ ⎩ E2 (s) = s . sq2 + k

From the final-value theorem of the Laplace transform[1–2] , we have (1)

lim e1 (t) = lim+ sE1 (s) = lim+

sq1 e1 (0) + sq1 −1 e1 (0) = 0, sq1 + 1

lim e2 (t) = lim sE2 (s) = lim

sq2 e2 (0) + sq2 −1 e2 (0) = 0. sq2 + k

t→∞

t→∞

s→0

s→0+

s→0

s→0+

(1)

Therefore, the synchronization of the drive system (23) and the response system (22) can be achieved under the control law (28). To verify and demonstrate the effectiveness and the feasibility of the presented synchronization method, we will give the simulation results including the synchronization of commensurate order and incommensurate order Duffing systems. For convenience, the parameters of Eq. (19) are specified as k = 0.5, a = b = 1, and ω = 1 throughout the paper. The initial conditions


575

for the drive system (22) and the response system (23) are x1 (0) = 0, y1 (0) = 0 and x2 (0) = 2, y2 (0) = 3, respectively. Thus, the initial conditions for the error systems are e1 (0) = 2,

e2 (0) = 3.

Figure 1 plots the phase portraits of Eqs. (22) and (23) in which q1 , q2 = 0.95 and r = 1. Figures 2 and 3 show the synchronization of the commensurate order system in which q1 , q2 = 0.95 and r = 1. Figure 4 depicts the phase portraits of Eqs. (22) and (23) with q1 = 0.95, q2 = 0.98, and r = 0.95. Figures 5 and 6 display the synchronization of the incommensurate order system in which q1 = 0.95, q2 = 0.98, and r = 0.95. Figures 3 and 6 display that the error dynamic system (25) quickly settles to zero, which implies chaos synchronization.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Phase portraits of Eqs. (22) and (23) with q1 , q2 = 0.95 and r = 1

Time series of Eqs. (22) and (23) with q1 , q2 = 0.95 and r = 1

Evolution of error functions for Eqs. (22) and (23) with q1 , q2 = 0.95 and r = 1

Phase portraits of Eqs. (22) and (23) with q1 = 0.95, q2 = 0.98, and r = 0.95

576


Fig. 5

Fig. 6

Time series of Eqs. (22) and (23) with q1 = 0.95, q2 = 0.98, and r = 0.95

Evolution of error functions for Eqs. (22) and (23) with q1 = 0.95, q2 = 0.98, and r = 0.95

5.2 Synchronization of fractional order Duffing system with single control function In this subsection, the goal is to study the chaos synchronization of fractional order Duffing systems by using single active control. The drive and response systems are given as follows: q1 D∗ x1 = y1 , (30) D∗q2 y1 = −ky1 + ax1 − bx31 + r cos(ωt), q1 D∗ x2 = y2 , (31) D∗q2 y2 = −ky2 + ax2 − bx32 + r cos(ωt) + v2 . The error variables are defined as Eq. (21). By subtracting Eq. (30) from Eq. (31) and using Eq. (24), we obtain

D∗q1 e1 = e2 , D∗q2 e2 = −ke2 + ae1 − b(x32 − x31 ) + v2 .

(32)

Theorem 4 Systems (30) and (31) will approach global synchronization for any initial conditions with the following control laws: v2 = −(1 + a)e1 + b(x32 − x31 )

(33)

if

k 2 or

is satisfied.

⎧ ⎪ ⎨ 0 < k < 2, √ 2 ⎪ ⎩ 4 − k > tan π γ k 2

(34)


Proof The substitution of Eq. (33) into Eq. (32) yields q1 D∗ e 1 = e 2 , D∗q2 e2 = −e1 − ke2 .

577

(35)

And its corresponding characteristic equation is λ2 + kλ + 1 = 0. If k 2, the corresponding eigenvalues are √ −k ± k 2 − 4 < 0. λ1,2 = 2 According to Proposition 2, it is direct to see that Eq. (35) is globally asymptotically stable if Eq. (34) is satisfied. Therefore, the synchronization of the drive system (30) and the response system (31) is achieved. If k < 2, the corresponding eigenvalues are √ −k ± i 4 − k 2 λ1,2 = . 2 Then, π |arg(λ1,2 )| > γ 2

if

√ π 4 − k2 > tan γ . k 2

According to Proposition 2, it is direct to see that Eq. (35) is globally asymptotically stable if Eq. (34) is satisfied. Therefore, the synchronization of the drive system (30) and the response system (31) is achieved. This completes the proof. The effectiveness of Theorem 4 can be demonstrated through the following numerical simulation. The parameters of the fractional order Duffing system are taken as k = 0.5 and a, b, ω = 1. The initial conditions for the drive and response systems are x1 (0) = 0, y1 (0) = 0 and x2 (0) = 2, y2 (0) = 3, respectively. (i) Commensurate case (q1 , q2 = 0.95 and r = 1) Figures 7–9 plot the phase portraits and synchronization of Eqs. (30) and (31) with

Fig. 7

Fig. 8

Phase portraits of Eqs. (30) and (31) with q1 , q2 = 0.95 and r = 1

Time series of Eqs. (30) and (31) with q1 , q2 = 0.95 and r = 1

578


Fig. 9

Evolution of error functions for Eqs. (30) and (31) with q1 , q2 = 0.95 and r = 1

q1 , q2 = 0.95 and r = 1. It is seen that the master system (30) and the slave system (31) tends to synchronization as time tends to infinity. (ii) Incommensurate case (q1 = 0.95, q2 = 0.98, and r = 0.95) Figure 10 depicts the phase portraits of Eqs. (30) and (31) with q1 = 0.95, q2 = 0.98, and r = 0.95. Figures 11 and 12 show the synchronization of the commensurate order Duffing system. It is seen that the error dynamic system (32) quickly settles to zero as time tends to infinty.

Fig. 10

Fig. 11

Fig. 12

Phase portraits of Eqs. (30) and (31) with q1 = 0.95, q2 = 0.98, and r = 0.95

Time series of Eqs. (30) and (31) with q1 = 0.95, q2 = 0.98, and r = 0.95

Evolution of error functions for Eqs. (30) and (31) with q1 = 0.95, q2 = 0.98, and r = 0.95


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5.3

Synchronization between fractional order Duffing and van der Pol-Duffing systems In this subsection, we study the synchronization between the fractional order Duffing system and the van der Pol-Duffing system. Assume that the fractional order van der Pol-Duffing system is synchronized with the fractional Duffing system. Define the drive system as q1 D∗ x1 = y1 , (36) D∗q2 y1 = μ(1 − x21 )y1 + a ˜x1 − ˜bx31 + R cos(Ωt)

and the response system as q1 D∗ x2 = y2 + u1 , D∗q2 y2 = −ky2 + ax2 − bx32 + r cos(ωt) + u2 .

(37)

The error variables are defined as Eq.(24). For synchronization, it is essential that the errors ei → 0 (i = 1, 2) as t → ∞. Note that q1 D∗ e 1 = e 2 + u 1 , (38) D∗q2 e2 = −ky2 + ax2 − bx32 + r cos(ωt) − μ(1 − x21 )y1 − a ˜x1 + ˜bx31 − R cos(Ωt) + u2 . The control functions ui are defined as u1 = V1 (t), u2 = (k + μ)y1 − μx21 y1 − (a − a ˜)x1 + bx32 − ˜bx31 − r cos(ωt) + R cos(Ωt) + V2 (t).

(39)

The terms Vi (t) are linear functions of the error terms ei (t). With the choice of ui (t) given by Eq. (39), the error system (38) becomes q1 D∗ e1 = e2 + V1 (t), (40) D∗q2 e2 = −ke2 + ae1 + V2 (t). The control terms Vi (t) are chosen so that the system (40) becomes stable. There is not a unique choice for such functions. We choose V1 e1 =A , (41) V2 e2 where A is a 2 × 2 real matrix, satisfying that for all eigenvalues λi of the system (40), | arg(λi )| > If we choose

A=

−1 −a

π γ. 2

−1 k−1

(42) ,

(43)

then the eigenvalues of the linear system (10) are −1 and −1. Therefore, the system (40) is stable, and the synchronization of Eqs. (36) and (37) is achieved. The parameters of the fractional order Duffing system are taken as k = 0.5,

a = b = ω = 1,

r = 0.95,

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and the parameters of the fractional order van der Pol-Duffing system are μ = 0.1,

a ˜ = ˜b = 0.5,

R = 0.75,

Ω = 2.

The fractional orders are taken to be q1 = 0.96 and q2 = 0.98. The initial conditions for the systems (36) and (37) are x1 (0) = 0, y1 (0) = 0 and x2 (0) = 2, y2 (0) = 3, respectively. For the error system (40), the initial conditions turn out to be e1 (0) = 2 and e2 (0) = 3. Figure 13 plots the phase portraits of Eqs. (36) and (37). The synchronization simulation results are summarized in Figs. 14 and 15. It is clearly shown that the synchronization is achieved as time tends to infinity. Figure 15 shows that the evolution of all the variables of the error dynamic system (40) quickly settle to zero, which implies chaos synchronization.

Fig. 13

Fig. 14

Fig. 15

Phase portraits of Eqs. (36) and (37)

Time series of Eqs. (36) and (37)

Evolution of error functions for Eqs. (36) and (37)

5.4 Potential application in secure communication The idea to exploit fractional order chaos systems in communication applications has been sparked when scientists and engineers are looking for practical applications. The general structure of chaos synchronization communication schemes is shown in Fig. 16. One can classify three potential application fields, which follow from three different behavior aspects as follows.


581

(i) Broad-band aspect Chaotic signals are inherently nonperiodic and possess a continuous spectrum. In communications, broad-band signals are used to fight channel imperfections. Therefore, fractional chaotic signals become candidates for spread-spectrum communications. (ii) Complexity aspect Fractional order chaotic systems have larger key space than integer order chaotic systems because they have much more parameters (fractional orders can be regarded as parameters). Moreover, fractional chaotic signals have a complex structure. This makes it difficult to guess the structure of the generator and to predict the signals over long time intervals. (iii) Orthogonality aspect Fractional chaotic signals are aperiodic, and thus have a vanishing autocorrelation function. The orthogonality can be exploited in multiuser communication applications.

Fig. 16

6

Structure of fractional chaos communication scheme

Conclusions

In the present work, the dynamics for the fractional order Duffing system is extensively investigated. The existence and uniqueness of the solutions for the fractional order Duffing system and its stability of equilibria are provided. The synchronization of fractional non-autonomous systems is investigated by the stability criterion of linear fractional systems. Especially, the single active control technique is applied to synchronize the fractional order Duffing system. The corresponding numerical simulations demonstrate the effectiveness of the proposed synchronization techniques. The work is helpful for the theoretical investigation of fractional nonlinear oscillators, and suggests some potential applications in secure communication.

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