Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system

Optik 130 (2017) 189–200

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Optik journal homepage: www.elsevier.de/ijleo

Original research article

Hopf bifurcation and chaos in a fractional order delayed memristor-based chaotic circuit system Wei Hu, Dawei Ding ∗ , Yaqin Zhang, Nian Wang, Dong Liang School of Electronics and Information Engineering, Anhui University, Hefei 230601, China

a r t i c l e

i n f o

Article history: Received 3 September 2016 Accepted 30 October 2016 Keywords: Hopf bifurcation Chaos Stability Time delay Fractional order Memristor

a b s t r a c t This paper present Hopf bifurcation and chaos in a fractional order delayed memristorbased chaotic circuit system. Firstly, regarding the time delay  as a bifurcation parameter, we investigate the stability and bifurcation behaviors of this fractional order delayed memristor-based chaotic circuit system. Some explicit conditions for describing the stability interval and emergence of Hopf bifurcation are derived. Secondly, corresponding to different system parameters, the complex dynamics behaviors of this system are discussed by using the bifurcation diagrams, the Max Lyapunov exponents (MLEs) diagram, the time domain waveforms, the phase portraits and the power spectrums. Thirdly, we study the influence of the two parameters (time delay  and fractional order q) on the chaotic behavior, and it is found when time delay  and fractional order q increases, the transitions from period one to period two and period four to chaos are observed in this memristor–based system. Meanwhile, corresponding critical values of time delay  and fractional order q, the lowest orders q and the minimum time delay  for generating chaos in the fractional order delayed memristor–based system are determined, respectively. Also, when the system occurs period one, the corresponding frequency is verified theoretically and experimentally. Finally, numerical simulations are provided to demonstrate the validity of theoretical analysis using the improved Adams–Bashforth–Moulton algorithm. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction In 1971, according to the completeness of axiomatic system in circuit theory, Professor Leon O. Chua defined the relationship between the magnetic flux and charge, and he put forward the concept of a memristor which was a fourth basic circuit components (the resistor, inductor, capacitor) [1–5]. In May 2008, the Stanley Williams team of HP Labs successfully developed a memristor physical model, which can work on confirming the existence of the memristor [6]. Since then, the memristor has been development rapidly, which has aroused widespread concern in the academia and industry. Because of a nonlinear characteristic in the memristor, it has great potential application such as reconfigurable logic and programmable logic devices, neural network [7–9], signal processing [10,11], non-volatile memory [12] and non-linear circuit [13–16] and so on. Accordingly, with the increasing depth research of the theoretical model and device performance in memristor, a variety of memristor mathematical models have been proposed and used in various applications. Also, using the memristor in secure communications is a relatively important area in the field of artificial intelligence, which attracts many researchers

∗ Corresponding author. E-mail address: [email protected] (D. Ding). http://dx.doi.org/10.1016/j.ijleo.2016.10.123 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

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to work. This is a very wide range of applications that was used to build a memristor circuit to generate non-linear chaotic signal to the field in secure communications. Fractional calculus is an extension and generalization of the integer order calculus to any real order of calculus. It has more than 300 years of history [17,18]. Because of no real prospect of the theory in fractional calculus a long time, it has not developed slowly. Comparing to an integer-order differential equations, fractional differential equations can more accurately describe many natural phenomena in many fields of applied science, such as electromagnetic waves, polarization electrolyte viscoelastic system, encryption [19], biology, system control [20–22], signal processing [23] and so on. However, it is the first time that the Mandelbrot pointed out there are a lot of fractal dimension theory in many fields of science and technology in 1983 [24]. Then, the application of fractional calculus is caused widespread concern, and gradually become a hot research topic of nonlinear theory. Fractional order system study is based to integer order system study, that fractional differential operator is introduced into the nonlinear dynamic system in the integer-order system. Then, many more classic dynamics behaviors of fractional order system are analyzed, such as Chua’s circuit [25], Duffing system [26], Chen system [27] and so on. Then, it is inevitable that the existence of time delay in many systems. Although the time delay is very small, it often affects the dynamic behavior of the whole system. Therefore, the delayed fractional order system must be considered to study. Therefore, a large number of researchers studied the fractional order delayed system, such as economy [28], biology [29], system control [30–32] and so on. As far as we know, there are many investigations on the complex dynamics analysis of bifurcation and chaos in the fractional order delayed system [33–36]. In [33], the fractional order delayed predator-prey systems with Holling type-II functional response was analyzed, and Hopf bifurcation occured when time delay  passed through a sequence of critical values. In [34], the bifurcation analysis of a fractional order single cell with delay which was proposed for delayed cellular neural networks with respect to the time delay . In [35], the dynamic of a tri-neuron fractional neural network was investigated, and applying the sum of time delay as the bifurcation parameter was studied. Also, Dumitru et al. [36] analyzed chaotic behavior in the fractional order nonlinear Bloch equation with delay. However, those delayed system is only study one of the bifurcation analysis and chaos, and there are few results with the respect to the system that is analyzed of the dynamic in Hopf bifurcation and chaos behavior. Till now, there are few results with the Hopf bifurcation and chaos behavior in a fractional order delayed memristorbased chaotic systems. Donato et al. [37] was apply the integer order memristor-based chaotic system to the fractional order system, and chaotic behaviors was shown. Lin et al. [38] also analyzed the chaotic behavior in a fractional order memristorbased chaotic circuit using the fourth degree polynomial. MS et al. [39] analyzed the Hopf bifurcation and chaos in a fractional order memristor-based electrical circuit in system parameter and fractional order q. However, those memristor-based chaotic systems are all not considered the exist of time delay . So it is necessary and challenging that we study two parameters of time delay  and fractional order q to influence complex dynamic behaviors in the memristor-based circuit system. Through the above discussion, this paper presents a system based on delayed memristor circuit to analysis the Hopf bifurcation and chaotic behavior. The main contribution of this paper is to present the influence and relation of the two parameters of the fractional order q and the time delay  on the dynamic behavior of the system. Accordingly, nonlinear dynamic is illustrated using time-domain diagram, phase diagram, spectrum, bifurcation diagram, Max Lyapunov exponents. Simulation results show that the fractional order delayed systems can generate complex nonlinear dynamic behavior with lower order. Also, the bifurcation point of time delay  ∗ and fractional order q∗ is to verify correctly in the theoretical value and the critical frequency in period one is verified accurately. This article follows: The Section 2 describes the concept of memristor, definition of the fractional calculus. The fractional order delayed memristor-based chaotic circuit system is investigated. The Section 3 focus on analysis of stability and Hopf bifurcation analysis in the memristor-based circuit system. The Section 4 presents the Hopf bifurcation and chaos dynamic in the fractional order delayed memristor-based chaotic circuit. The Section 5 concludes the paper.

2. Background 2.1. Memristors A memristor is a circuital element to show the relationship in the charge (z) and the flux (). There are two forms of the memristor. Firstly, the charge-controlled memristor is defined by

⎧ ⎪ ⎨ vM = M (z) iM , ⎪ ⎩ M (z) = d (z) , dz

where M(z) is the memristance.

(1)

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Fig. 1. The system in a delayed Memristor-based chaotic circuit.

Secondly, the flux-controlled memristor is defined by

⎧ ⎪ ⎨ iM = W () vM ,

(2)

⎪ ⎩ W () = dz () , d

where W () is the memductance, and vM ,iM , z and  are the voltage, current, charge and flux across the memristor. 2.2. Fractional calculus In order to discuss the fractional order chaotic system, we often need to solve fractional differential equations. Three commonly definitions for fractional derivatives [40] have been proposed such as Riemann–Liouville, Grünwald–Letnikov and Caputo definitions. Definition 1.

([40]):The Caputo fractional derivative definition of fractional order q of non-integer is 1  (m − q)

q

Dt f (t) =



t

0

f (m) () (t − )q+1−m

d,

m − 1 < q < m,

(3)

where  is Gamma −function, and



∞

 (z) =

e−t t z−1 dt,

(4)

0

 (z + 1) = z (z). Definition 2.



q

(5)

([41]):The Laplace transform of Caputo derivative is



L Dt f (t)



m−1

= sq F(s) −

sq−1−k f (k) (0), m − 1 < q < m,

(6)

k=0

where Laplace transform of f (t),f k (0), k = 1, 2, . . ., n, are the initial condition. If f k (0) = 0, k = 1, 2, . . ., n, then  q F(s)  is the q L Dt f (t) = s F(s). 2.3. The system in the delayed memristor-based chaotic circuit In this section, the delayed Memristor-based chaotic circuit system which is shown in Fig. 5 [42] has been introduced. The circuit system consists of a nonlinear active delayed memristor-based system and a capacitor. This memristor-based chaotic circuit system in Fig. 1 is controlled a port by the change of the system voltage, and the following equations describe x˙ = f (x , vM , t) = ax + b|x | + c vM , iM = G(x, vM , t)vM = (˛ + ˇx)vM .

(7)

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We apply for the Kirchhoff’s circuit laws to the system in Fig. 1, and we can obtain the following circuit equations dx (t) = ax (t − ) + b|x (t − ␶) | + c vc (t) , dt

(8)

dvC (t) C = −˛vC (t) − ˇx (t) vC (t) , dt where ␶ is the time-delay, x is the state variable of the memristor, and a, b, c, ˛, ˇ are constants. As follows, the dimensionless equations are deduced x˙ = ax + b|x | + cy,

(9)

y˙ = my + nxy, where y = vC (t),x = x(t − ),m = −˛/C and n = −1/RC. The models of the delayed fractional order memristor-based circuit system given in q

Dt 1 x = ax (t − ) + b|x (t − ) | + cy, q

Dt 2 y = my + nxy,

(10)

where q1 ,q2 are the fractional order of the electrical elements: memristor M and capacitor C. 3. Stability analysis and Hopf Bifurcation analysis In this section, we will discuss the stability and Hopf bifurcation of the delayed fractional order memristor-based circuit system(10) with fractional order q and time delay . 3.1. Stability analysis For the selected system parameters, there are two real equilibrium points X ∗ = (x∗ , y∗ )of the system (10) viz.X1 ∗ = (0, 0) m(a+b) ∗ m m(a+b) and X2 ∗ = (− m n, nc ).In the following,we only discuss the equilibrium X2 = (− n , nc ). ∗ ∗ Assume that X(t) = x(t) − x , Y (t) = y(t) − y , the linearized system of nonlinear system near an equilibrium point X2 ∗ is q

Dt X = JX + J X ,

(11)

where X and X are column vectors (x, y) at t and at t − , respectively, and



J= and

0 m(a + b) c



J =

c



0

,

(12)

a + b ∗ sign(x∗ )

0

0

0

 ,

(13)

are Jacobians with respect to X and X , respectively. If x∗ > 0,we can get linearized system is

⎧ q ⎪ ⎨ Dt X = (a + b) X + cY,

(14)

⎪ ⎩ Dq Y = m (a + b) X. t c

The characteristic equation is sq − (a + b ∗ sign(x∗ ))e−s |

−

m(a + b) c

−c sq

| = 0.

(15)

We can get s2q + P1 e−s sq + P2 = 0,

(16)

where P1 = −(a + b ∗ sign(x∗ )), P2 = −m(a + b). Suppose that s = ωi = |ω|(cos 2 + i sin(± 2 )) is a root of Eq. (16), where ω is real number and when ω > 0, we can get i sin 2 , while when ω < 0,we can get −i sin 2 .

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Separating real and imaginary parts gives



⎧ q q q 2q q ⎪ ⎨ |ω| cos q + P1 |ω| cos 2 cos ω + |ω| sin(± 2 ) sin ω + P2 = 0,

⎪ ⎩ |ω|2q sin(±q) + P1 −|ω|q cos q sin ω + |ω|q sin(± q ) cos ω = 0. 2

(17)

2

We can suppose that



A1 + B cos ω + C sin ω = 0,

(18)

A2 − B sin ω + C cos ω = 0. where

⎧ A1 = |ω|2q cos q + P2 , ⎪ ⎪ ⎪ ⎪ ⎪ 2q ⎪ ⎨ A2 = |ω| sin(±q), (19)

q

, B = P1 |ω|q cos ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ C = P1 |ω|q sin(± q ). 2

We can get B2 + C 2 = A21 + A22 ,

(20)

Then we can obtain |ω|4q + ˚1 |ω|2q + ˚2 = 0,

(21) 2 + (a + b ∗ sign(x∗ )) ), ˚2

where ˚1 = −(2m(a + b) cos q Label h(ω) = |ω|4q + ˚1 |ω|2q + ˚2 . Let

=

2 m2 (a + b) .

(H1 )P1 = −(a + b ∗ sign(x∗ )), (H2 )P2 = −m(a + b), 2

(H3 )˚1 = −(2m(a + b) cos q + (a + b ∗ sign(x∗ )) ), 2

(H4 )˚2 = m2 (a + b) > 0. In the following, the Lemma is imperative to be raised to discuss the distribution of the roots of Eq. (16). Lemma 1.

For Eq. (16), the results hold:

/ 0, then Eq. (16) has no root with zero real parts for all  ≥ 0. i) If (H3 )˚1 > 0, (H4 )˚2 > 0,(H2 )P2 = ii) If (H3 )˚1 < 0,(H4 )˚2 > 0, then Eq. (16) has a pair of purely imaginary roots ± ω0 when  = j , j = 0, 1, 2, · · ·, where (j)

k =





1 A1 B + A2 C arccos − 2 ωk B + C2



+ 2j , j = 0, 1, 2· · ·, k = 1, 2, 3...

(22)

where ω0 is the unique positive zero of the function h(ω). i) From (H3 )˚1 > 0, (H4 )˚2 > 0, we can derive h(0) = ˚2 > 0, and

Proof. 

h (ω) = 4q|ω|4q−1 + 2q˚1 |ω|2q−1 . Combine q > 0 and(H3 )˚1 > 0, (H4 )˚2 > 0, we claim that Eq. (21) has no real root, and hence Eq. (16) has no purely imaginary root. Provided that P 2 = / 0,s = 0 is not a root of Eq. (16).  ii) By means of (H4 )˚2 > 0, it is easy to conclude that h(0) = ˚2 > 0. Then, by lim h(ω) = +∞, and h (ω) = 4q|ω|4q−1 + ω→+∞

2q˚1 |ω|2q−1 < 0 for ω > 0, there exists a unique positive number ω0 such that h(ω0 ) = 0. Then ω0 is a root of Eq. (21). Hence, for j as defined in Eq. (22), (ω0 , j ) is a root of Eq. (17). It can be seen that ±ω0 is a pair of purely imaginary roots of Eq. (16) when  = j , j = 0, 1, 2· · ·..

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3.2. Hopf bifurcation analysis In the following, the hypothesis is proposed to derive the condition of the occurrence for Hopf bifurcation. (H5 ).

dRe(s) 1 3 + 2 4 = / 0. = d 32 − 42

where i , i = 1, 2, 3, 4 are defined by Eq. (25). Lemma 2. Let s() = () + iω() be the root of system Eq. (16) near  = j satisfying (j ) = 0, ω(j ) = ω0 the transversality condition holds

Re

Proof.

ds d

|(=0, ω=ω0 ) = / 0.

Differentiating Eq. (16) implicitly with respect to,we obtain −(a + b ∗ sign(x∗ ))e−s sq+1 ds , = 2q−1 d 2qs + (a + b ∗ sign(x∗ ))sq − (a + b ∗ sign(x∗ ))qe−s sq−1

=

2qs2q−1

P1 e−s sq+1 . − P1 e−s sq + P1 qe−s sq−1

(23)

Therefore, when s = iω,  = 0 , Eq. (23) can become as follows: 1 + i2 ds = , d 3 + i4

(24)

where

⎧ (q + 1) (q + 1) ⎪ q+1 ⎪ + sin ω sin ], ⎪ 1 = P1 ω [cos ω cos 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ (q + 1) (q + 1) ⎪ ⎪ − sin ω cos ], 2 = P1 ωq+1 [cos ω sin ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 3 = 2qω2q−1 cos (2q − 1) − P1 ωq [cos ω cos q + sin ω sin q ] 2

2

2

⎪ ⎪ (q − 1) (q − 1) ⎪ ⎪ + sin ω sin ], +P1 qωq−1 [cos ω cos ⎪ ⎪ 2 2 ⎪ ⎪ (2q − 1) q q q ⎪ 2q−1 4 = 2qω − P1 ω [cos ω sin − sin ω cos ] sin ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ +P1 qωq−1 [cos ω sin (q − 1) − sin ω cos (q − 1) ]. 2

(25)

2

So dRe(s) 1 3 + 2 4 = , d 32 − 42

(26)

and dIm(s) 2 3 − 1 4 . = d 32 − 42 Apparently, the hypothesis (H5 ) implies that transversality condition is satisfied. The proof is complete. Though the Routh-Hurwitz, we can obtain 1 = P1 , 2 = |

Theorem 1.

P1

0

1

P2

| = P1 P2 .

For this delayed fractional order memristor-based linear system (10), the following results hold

(27)

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Table 1  ds  |(=0, ω=ω0 ) = / 0 of this system (10) for different fractional order q. Critical frequency ω0 , Critical frequency f0 , bifurcation point0 and Re d



Re Fractional order q

Critical frequency ω0 (rad/s)

Critical frequency f 0 (Hz)

Bifurcation point 0

1 0.95 0.9 0.85 0.8 0.75 0.7 0.672 0.65 0.6

1.3660 1.3834 1.3918 1.3889 1.3718 1.3366 1.2785 1.2335 1.1907 1.0637

0.2174 0.2202 0.2215 0.2211 0.2183 0.2127 0.2035 0.1963 0.1895 0.1693

1.1499 1.0371 1.3241 1.4248 1.5420 1.6847 1.8690 2.0004 2.1244 2.5150

ds d

|(=0, ω=ω0 ) = / 0

−6.0685 −1.7246 −0.8627 −0.5107 −0.3274 −0.2188 −0.1488 −0.1199 −0.1008 −0.0652

i) If (H3 )˚1 > 0, (H4 )˚2 > 0 and i > 0, i = 1, 2, the positive equilibrium (x∗ , y∗ ) of this system (10) is locally asymptotically stable. ii) If (H3 )˚1 < 0, (H4 )˚2 > 0 and i > 0, i = 1, 2, the positive equilibrium (x∗ , y∗ ) of this system (10) is unstable. iii) If all the conditions as stated in (ii) hold, this system (10) undergoes a Hopf bifurcation at (x∗ , y∗ ) when  = j , j = 0, 1, 2· · ·, where  = j as defined in Eq. (22). Proof.

When  = 0, the characteristic equation (16) have 2

+ P1 + P2 = 0. i) According to the Routh–Hurwitz criterion, all the roots of Eq. (16) have negative real parts if and only if i > 0, i = 1, 2. Hence, all roots of Eq. (16) with  = 0 have negative real parts. The conclusion (i) of Lemma 1 indicates that Eq. (16) has no root with zero real part for all  > 0. Thus, all the roots of Eq. (16) have negative real parts for all  ≥ 0. ii) From the conclusion ii) of Lemma 1, the definition of 0 implies that all the roots of Eq. (16) have negative real parts for  ∈ [0, 0 ). The conclusion in Lemma 2 indicates that Eq. (16) has at least a couple of roots with positive real parts when  > 0 . iii) The conclusion in Lemma 2 implies that the transversality condition for Hopf bifurcations is satisfied under the given (j) assumption. So the Hopf bifurcation occurs at  = k , j = 0, 1, 2, .... 4. Numerical results In this section, the dynamical behaviors of system (10) are investigated numerically by means of bifurcation diagram, and the Max Lyapunov exponents, time-domain diagram, phase diagram, power spectrums. In this paper, the improved Adams-Bashforth-Moulton predictor algorithm [43] is utilized to solve fractional differential equations. Then, in order to investigate chaotic behaviors, the algorithm of the Max Lyapunov exponents is proposed by Wolf [44]. Accordingly, as we all know, the Max Lyapunov exponents increase from negative number to zero when periodic cycles appear. Also, the chaotic dynamics occurs when WLEs is positive. Finally, we give the system (10) parameter a = 1, b = −2, c = 5, m = 0.5, n = −0.9.By calculating, it is easy to obtain that the equilibrium points X1 ∗ = (0, 0) and X2 ∗ = (5/9, 1/9). In the following, we only discuss the equlilibrium X 2 ∗ . Neverthe/ 0), ˚2 = 0.25, i > 0, i = 1, 2,(P1 = 1, P2 = 0.5) is satisfied Theorem 1 ii). In there, we less, the ˚1 = cos q − 1 < 0(q = change the fractional order q ∈ [0.6,  1], and we can calculate the corresponding the bifurcation point , frequency ω/f and the transversality condition Re ds/d |(=0, ω=ω0 ) , which is shown in Table 1. We also can find the relationship in Figs. 2 and 3. In the following, numerical simulations show the strong effect of the fractional order parameters q and time delay  with its bifurcation point  ∗ = 1.3241 and q∗ = 0.672 on the dynamic of system (10), as shown in Fig. 2(b). We show some attractions of the system (10) like complex behaviors such as the fixed points, periodic orbits and strange attractors. Table 2 is concluding the detailed dynamical behaviors of this system (10) with different fractional order q or different time delay . 4.1. Bifurcation and chaos versus the time delay  Firstly, fixing the parameter of the fractional order (q = 0.9) and varying the parameter of the time delay ∈ [1, 2.5], the bifurcation diagrams are shown in Fig. 4(a) and the Max Lyapunov exponent corresponding to Fig. 4(a) is given in Fig. 5(a). By calculating, the critical bifurcation point  ∗ = 1.3241.The transversality condition (H5 ) is satisfied. Form Theorem 1, the equilibrium(x∗ , y∗ ) is stable and unstable if time delay 1 = 1.30 < 0 = 1.3241 and 2 = 1.34 > 0 = 1.3241, which is shown Fig. 6(a) and Fig. 6(b), respectively. When  = 1.3241,the system occurs in the a Hopf bifurcation. The transitions

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Fig. 2. (a) Critical frequency ω0 /f0 of different fractional order q ∈ (0.6, 1). (b)Bifurcation point 0 of different fractional order q ∈ (0.6, 1).

Fig. 3. Evolution of Re

 ds  d

|(=0, ω=ω0 ) and critical frequency ω0 /f0 versus (q, 0 ).

from period one to period two and period four to chaos are observed in this memristor–based system. The fixed point, the period one, period two, period four and the strange attractors occurs in Fig. 8(a), Fig. 8(b), Fig. 8(c), Fig. 8(d) and Fig. 8(e), respectively. When the period one occurs, the theoretical calculations obtained the frequency f = 0.2215Hz is similar to the actual calculated value, which can see Table 1 and Fig. 8(b3 ). When(b3 )·When the fractional order  ∈ [1.39, 1.61],the period

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Table 2 The detailed dynamical behaviors of this system (10) with different fractional orders or different time delay . Fractional order q

Time delay 

Behavior

Figure

0.9

1.2 1.35 1.5 1.63 2.2

Fixed point Period-1 Period-2 Period-4 Chaos

8(a) 8(b) 8(c) 8(d) 8(e)

0.62 0.7 0.75 0.77 0.8 0.82 0.85

2

Fixed point Period-1 Period-2 Period-4 Chaos Period-3 Chaos

9(a) 9(b) 9(c) 9(d) 9(e) 9(f) 9(g)

Fig. 4. (a)Bifurcation diagram x(t) of q = 0.9, (b)Bifurcation diagram x(t) of  = 2..

Fig. 5. (a) Max Lyapunov exponents max of q = 0.9 (b) Max Lyapunov exponents max of  = 2.

Fig. 6. Time domain waveforms and phase portrait of the system with 1 = 1.30 < 0 = 1.3241 of the initial values(0.1,0.1), and 2 = 1.34 > 0 = 1.3241 of the initial values (1.5, 0.5) in the fractional order q = 0.9.

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Fig. 7. Time domain waveforms and phase portrait of the system with q1 = 0.66 < q0 = 0.672 of the initial values (0.1,0.1), q2 = 0.68 > q0 = 0.672 of the initial values (0.1, 0.1) in the time delay  = 2.

Fig. 8. The time-domain diagram, phase diagram and power spectrums of the systems with different time delay when the fractional order q = 0.9. (a1 )The time-domain diagram with  = 1.2. (a2 ) phase diagram with  = 1.2. (a3 ) power spectrums with  = 1.2. (b1 )The time-domain diagram with  = 1.35. (b2 ) phase diagram with  = 1.35. (b3 ) power spectrums with  = 1.35. (c1 )The time-domain diagram with  = 1.5.(c2 ) phase diagram with  = 1.5. (c3 ) power spectrums with  = 1.5. (d1 )The time-domain diagram with  = 1.63.(d2 ) phase diagram with  = 1.63. (d3 ) power spectrums with  = 1.63. (e1 )The time-domain diagram with  = 2.2. (e2 ) phase diagram with  = 2.2. (e3 ) power spectrums with  = 2.2.

two is appeared. When the fractional order  ∈ [1.61, 1.66], the period four is emerged. The chaos exists when decreasing the time dalay . When  ∈ [1.66, 2.5], the chaos appear, and the chaotic attractor of system(10) is presented in Fig. 8(e). 4.2. Bifurcation and chaos versus the fractional order q Secodly, we fix the time delay  = 2, and consider the fractional order q as a control parameter. From Fig. 4(b), the bifurcation diagram is depicted that decreasing the fractional order parameter q ∈ [0.6, 0.9]. The MLEs is shown in Fig. 5(b). By calculating, the critical bifurcation point q∗ = 0.672.The transversality condition (H5 ) is satisfied. Form Theorem 1, the equilibrium is stable or unstable if q1 = 0.66 < q0 = 0.672 or q2 = 0.68 > q0 = 0.672,which is shown Fig. 7(a) or Fig. 7(b), respectively. When q < 0.65, the system is locally asymptotically stable that is shown in Fig. 9(a). When q ≈ 0.65, the system occurs in the a Hopf bifurcation. With decreasing the fractional order q ∈ [0.65, 0.77], the period one to period two and to period four is appeared that is shown in Fig. 9(b), Fig. 9(c) and (d),respectively. In the same way, when system occurs period one, the frequency f = 0.1966Hz is checked in fractional order q = 0.69 theoretically and experimentally, which is seen in Table 1 and Fig. 9(b3 ). The chaotic behavior exists when decreasing the fractional order q. When q ∈ [0.77, 0.82], the chaos appear,

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Fig. 9. The time-domain diagram, phase diagram and power spectrums of the systems with different fractional order q when the time delay  = 2. (a1 )The time-domain diagram with q = 0.62. (a2 ) phase diagram with q = 0.62. (a3 ) power spectrums with q = 0.62. (b1 )The time-domain diagram with q = 0.69. (b2 ) phase diagram with q = 0.69. (b3 ) power spectrums with q = 0.69. (c1 )The time-domain diagram with q = 0.75. (c2 ) phase diagram with q = 0.75. (c3 ) power spectrums with q = 0.75. (d1 )The time-domain diagram with q = 0.77. (d2 ) phase diagram with q = 0.77. (d3 ) power spectrums with q = 0.77. (e1 )The time-domain diagram with q = 0.8. (e2 ) phase diagram with q = 0.8 . (e3 ) power spectrums with q = 0.8. (f1 )The time-domain diagram with q = 0.82. (f2 ) phase diagram with q = 0.82. (f3 ) power spectrums with q = 0.82. (g1 )The time-domain diagram with q = 0.85. (g2 ) phase diagram with q = 0.85. (g3 ) power spectrums with q = 0.85.

and the chaotic attractor of system(10) is presented in Fig. 9(e).When the fractional order q ∈ [0.82, 0.846], the period three is appeared, and it is shown in Fig. 9(f). The other chaotic attractor of system(10) is presented in Fig. 9(g) when the fractional order q ∈ [0.846, 0.9]. So, the fractional order delayed system exhibits rich dynamical behaviors and generate period, bifurcations and chaos in certain parameters regions.

5. Conclusion In this paper, we have introduced the fractional order delayed memristor-based chaotic circuit. We have also studied the stability and Hopf bifurcation in this system. The strange attractors of the system have been obtained. Moreover, it has been found that the fractional order q and time delay  have common effect on the stability of the system. We have also shown that when decreasing the parameter of the fractional orderqand increasing the time delay , the chaotic behavior of system (10) tend to be stabilized. It has also been shown that the fractional order delayed system exhibits much richer dynamical behaviors than its corresponding integer system. Numerical simulations have been used to show the bifurcation, chaotic attractors of the fractional order delayed system. In order to more efficient application in the field of secure communications, designing a specific hardware circuit model of the fractional order delayed memristor-based chaotic system will be considered in the near future.

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Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos: 61201227), Funding of China Scholarship Council, the Natural Science Foundation of Anhui Province (No: 1208085M F93), 211 Innovation Team of Anhui University (Nos: KJTD007A and KJTD001B). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

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