1 Introduction

Memristor, a type of two-terminal nonlinear circuital element, was put forward by Professor Chua in 1971^[1]. As a contemporary hot topic, the study of the nature of memristor has drawn wide attention from researchers. Compared with general differential systems, memristive differential systems with properties of memory, non-volatility, and nonlinearity may exhibit more complicated chaotic behaviors, which has potential application prospects. The current methods of building memristive circuit systems are no more than introducing the memristor in the existing circuit or replacing the nonlinear resistance element in the original circuit with memristor^[2-5].For instance, Ruan et al.^[2] introduced the simplified Lorenz system that uses memristor as a positive feedback and studied its dynamics. In Ref. [3] and Ref. [4], memristor was substituted for the resistor in the existing circuitry system to construct a new hyperchaotic memristive system with multi-winged attractors. On the contrary, many other scholars focused on the applications of memristor such as secure communications^[6], artificial neural networks^[7], and non-volatile random access memory^[8].

Fractional calculus has been highly favored in the study of chaos theory^[9-10] because it makes dynamic systems show more complex chaotic phenomena. As an exceptional case of system that is not linear, fractional-order chaotic systems have more complex dynamics than general nonlinear systems. It was a big hit when Hartley et al.^[11] first raised the Chua's system with fractional order in 1995. Then, in Ref. [12] a simple memristive chaos system with fractional order was studied, which is the extension of the integer order chaotic system. To sum up, the current research on fractional order is mostly based on the expansion of the corresponding system into its fractional order form^[13-15].

Time-delay (time interval) refers to the period from the initiation of a certain behavior to the results, which is frequently cited in the differential equations. Such delay differential dynamical systems are commonly used and studied in the fields of biological systems^[16], economy^[17], neural networks^[18], chaos^[19], physics^[20], etc. Delay seems to be very small, but sometimes it has a certain potential impact on the system's oscillation and may affect the dynamics of the entire system. Therefore, it is necessary to investigate time-delay in complex dynamic systems theoretically and practically.

Chaotic synchronization has always been a significant contemporary topic in the study of non-linear science. At present, there are various methods to realize synchronization of chaotic system schemes, such as Laplace transform theory^[21], adaptive feedback control method^[22], a sinusoidal state coupling control method^[23], active control method^[24], and sliding model control method^[25-27]. Some other methods were reported in Refs. [28-30]. In this paper, we derive sufficient criteria and design a linear feedback controller for system synchronization.

Motivated by what has been discussed, our research object is a fractional-order delayed memristive system without equilibrium. The main contributions are summarized as follows:

1) Taking the parameter a as bifurcation parameter, the evolution law of the proposed system with the change of parameter a was observed. It was found that this system exhibited more complicated chaotic behaviors than the integer order counterpart^[31];

2) A simple active control method was designed to cope with chaos synchronization. Then the stability of the error system was discussed to obtain the conditions for system synchronization;

3) The effects of three factors (i.e., parameter a, fractional order q, and time delay τ) on synchronization were discussed by providing examples and the results were analyzed.

The rest of this paper is arranged as follows. Section 2 gives the related definitions regarding to fractional calculus and proposes a delayed memristive chaotic differential system. Section 3 presents the evolution process with the variation of the coefficient parameter a. Section 4 introduces the process of chaos synchronization of the system and provides the stability conditions for the error system. The effects of these factors on the system synchronization are discussed in Section 5 and Section 6 summarizes the obtained findings.

2 Preliminaries 2.1 Definitions

Definition 1: Given that f(t) is a continuous function, the Caputo derivative adopting fractional order q is defined as

$ _0^CD_t^qf(t) = \frac{1}{{\Gamma (m - q)}}\int_0^t {\frac{{{f^{(q)}}(\tau )}}{{{{(t - \tau )}^{q - m + 1}}}}{\rm{d}}\tau } $

(1)

$ \Gamma (x) = \int_0^\infty {{t^{x - 1}}} {e^{ - t}}{\rm{d}}t $

(2)

where Γ is the Gamma function, 0≤m－1 < q≤m.

Definition 2: Laplace function defined by the Caputo definition is written as

$ L\left\{ {_0^CD_t^qf(t)} \right\} = {s^q}F(s) - \sum\limits_{k = 0}^{n - 1} {{s^{q - k - 1}}} {f^k}(0) $

(3)

where m－1 < q≤m.F(s) is obtained by the Laplace transform of f(t).

2.2 Fractional-Order Delayed Memristive System without Equilibrium Point

To introduce memristive systems, we generalize the original concept of the memristor and describe it as

$ \left\{ {\begin{array}{*{20}{l}} {n = W(m,u,t)u}\\ {\dot m = F(m,u,t)} \end{array}} \right. $

(4)

in which u, m, and n represent the internal state, input, and output of the memristive model.

The memristive model discussed in this paper is written as

$ \left\{ {\begin{array}{*{20}{l}} {h\left( {{x_1},{x_2}} \right) = \left( {0.1{x_2} - 1} \right){x_1}}\\ {{x_2} = c{x_1}} \end{array}} \right. $

(5)

Fig. 1 shows the circuit principle diagram of the discussed system^[31]. The delayed chaotic system consists of four resistors R₁, R₂, R₃, R₄, two integrators U₁, U₂, a delay unit, a capacitor C₁, a voltage follower U₃, DC voltage (V_c=1V_DC), and a generalized memory element.

By using the Kirchhoff's circuit laws, the characteristic equations of this system can be deduced as

$ \left\{ {\begin{array}{*{20}{l}} {\frac{{{\rm{d}}{x_1}}}{{{\rm{d}}t}} = - \frac{1}{{{R_1}{C_1}}}{x_1}(t - \tau ) + \frac{{{V_{{\rm{sat}}}}}}{{{R_2}{C_1}}}\text{sgn}\left( {{x_1}(t - \tau )} \right) - }\\ {\quad \frac{1}{{{R_3}{C_1}}}h\left( {{x_1},{x_2}} \right) - \frac{{{V_{\rm{C}}}}}{{{R_4}{C_1}}}}\\ {\frac{{{\rm{d}}{x_2}}}{{{\rm{d}}t}} = c{x_1}} \end{array}} \right. $

(6)

in which τ is the time delay, V_sat represents the saturation voltage of the operational amplifier U₂, and V_C=1V_DC.

Assume that

$ a = \frac{1}{{{R_1}{C_1}}},b = \frac{1}{{{R_3}{C_1}}} $

$ d = \frac{{{V_{\rm{C}}}}}{{{R_4}{C_1}}},\frac{{{V_{{\rm{sat}}}}}}{{{R_2}{C_1}}} = 1 $

$ h\left( {{x_1},{x_2}} \right) = \left( {0.1{x_2} - 1} \right){x_1} $

then the dimensionless form of Eq. (6) is written as

$ \left\{ {\begin{array}{*{20}{l}} {{{\dot x}_1} = - a{x_1}(t - \tau ) + sgn \left( {{x_1}(t - \tau )} \right) - }\\ {\quad bh\left( {{x_1},{x_2}} \right) - d}\\ {{{\dot x}_2} = c{x_1}} \end{array}} \right. $

(7)

Eq. (7) is generalized to fractional-order form and expressed as

$ \left\{ \begin{array}{l} {D^q}{x_1} = - a{x_1}(t - \tau ) + sgn \left( {{x_1}(t - \tau )} \right) - \\ \;\;\;\;\;\;\;bh\left( {{x_1},{x_2}} \right) - d\\ {D^q}{x_2} = c{x_1} \end{array} \right. $

(8)

in which 0 < q < 1, τ > 0, and a, b, c, d are positive constants.

$ \begin{array}{l} {\rm{Let}}\;{D^q}{x_1} = {D^q}{x_2} = 0,\text{then}\\ \left\{ {\begin{array}{*{20}{l}} { - a{x_1} + \text{sgn}\left( {{x_1}} \right) - b\left( {0.1{x_2} - 1} \right){x_1} - d = 0}\\ {c{x_1} = 0} \end{array}} \right. \end{array} $

(9)

Clearly, Eq. (9) has no real solution if d≠0, which means system (8) has no equilibrium point when d≠0. We also found that system (8) can exhibit chaotic attractors by numerical simulations although it has no equilibrium point. Therefore, the next step is to study the chaotic property of system (8) when there is no equilibrium.

3 Occurrence of Multiple Hidden Attractors in System (8)

At present, there are mainly three methods to solve the problem of chaotic systems with fractional order, namely, frequency domain method (FDM)^[32-33], Adams bashforth-moulton algorithm (ABM)^[34-35], and Adomian decomposition (ADM)^[36-37]. When detecting and analyzing chaotic behaviors in non-linear systems, it is relatively more precise and convenient for ABM and ADM, while FDM is not always useful and trustworthy. Thus we adopted the modified ABM in numerical simulations.

The simulation results of bifurcation diagram can be used to effectively determine the important dynamical properties of chaotic systems with fractional order. When fixing q=0.84, the parameters were set as b=0.02, c=0.1, d=0.001, τ=0.9. The initial conditions (0.1, 0.1) and the parameter a were adjusted. Then system (8) produced abundant phenomena with the variation of the parameter a (Fig. 2(a)). The asymptotically stable state of system (8) was observed in the interval a < 1.58. The stable state at a=1.5 is depicted in Fig. 3(a1)-(a2). Then Hopf bifurcation occurred and system (8) generated the limit cycle. Further, the increase of the parameter a led to the chaotic state of the system. The specific situation at a=1.795 is shown in Fig. 3(b1)-(b2). When a=1.9, the single scroll attractor appeared in system (8) as illustrated in Fig. 3(c1)-(c2). System (8) entered the hyperchaotic regime at a≥1.929 and the two-scroll hyperchaotic attractor of system (8) was generated at a≥1.969. The situation of the double hyperchaotic attractor for a=2.2 is depicted in Fig. 3(d1)-(d2).

When fixing q=0.9, system (8) exhibited a dynamic evolution law similar to q=0.84, from the asymptotically stable state to the appearance of the Hopf bifurcation to enter the chaotic area. Comparing Fig. 2(a) with Fig. 2(b), it was found that the larger the value of the order q was, the faster the evolution law accompanying the coefficient parameter a changed.

4 Synchronization of System (8)

System (8) is the drive system. By adding the controller to Eq. (8), the response system is rewritten as

$ \left\{ \begin{array}{l} {D^q}{y_1} = - a{y_1}\left( {t - \tau } \right) + \text{sgn}\left( {{y_1}\left( {t - \tau } \right)} \right) - \\ \;\;\;\;\;\;\;\;\;\;b\left( {0.1{y_2} - 1} \right){y_1} + {u_1}\left( t \right)\\ {D^q}{y_{\rm{2}}} = c{y_1} + {u_2}\left( t \right) \end{array} \right. $

(10)

in which u₁(t), u₂(t) are the appropriate control terms. The error of drive-response system can be set as

$ \begin{array}{*{20}{c}} {{e_1} = {y_1} - {x_1},{e_2} = {y_2} - {x_2}}\\ {{e_1}\left( {t - \tau } \right) = {y_1}\left( {t - \tau } \right) - {x_1}\left( {t - \tau } \right)} \end{array} $

(11)

For Eqs. (8)-(10), the following error system can be written as

$ \left\{ \begin{array}{l} {D^q}{e_1} = - a{e_1}(t - \tau ) + \text{sgn} \left( {{y_1}(t - \tau ) - } \right.\\ \;\;\;\;\;\;\;\text{sgn}\left( {{x_1}(t - \tau )} \right) + b{e_1} - 0.1b\left( {{x_1}{x_2} - {y_1}{y_2}} \right) + \\ \;\;\;\;\;\;\;{u_1}(t)\\ {D^q}{e_2} = c{e_1} + {u_2}(t) \end{array} \right. $

(12)

The nonlinear control input is defined as

$ \left\{ {\begin{array}{*{20}{l}} {{u_1} = \text{sgn} \left( {{x_1}(t - \tau )} \right) - \text{sgn} \left( {{y_1}(t - \tau )} \right) + }\\ {\;\;\;0.1b\left( {{x_1}{x_2} - {y_1}{y_2}} \right) - {K_1}{e_1}}\\ {{u_2} = - {K_2}{e_2}} \end{array}} \right. $

(13)

where K₁, K₂ are control parameters.

Substituting Eq. (13) into Eq.(12), then we obtain

$ \left\{ {\begin{array}{*{20}{l}} {{D^q}{e_1} = - a{e_1}(t - \tau ) + \left( {b - {K_1}} \right){e_1}}\\ {{D^q}{e_2} = c{e_1} - {K_2}{e_2}} \end{array}} \right. $

(14)

By discussing and analyzing the stability of error system (14) with zero solution, it is possible to reach synchronization between the drive system (8) and the response system (10). Then we studied the state of the positive and negative real parts of all the roots of the characteristic equation det(Δ(s)). If all of them possess negative real parts, the error system (14) with equilibrium point can be asymptotically stable^[38], and the antisense is not.

The characteristic equation is

$ \begin{array}{l} \text{det} (\mathit{\Delta} (s)) = \left| {\begin{array}{*{20}{l}} {{s^q} - \left( {b - {K_1}} \right) + a{{\rm{e}}^{ - s\tau }}}&0\\ { - c}&{{s^q} + {K_2}} \end{array}} \right| = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{s^{2q}} + \left( {{K_1} + {K_2} - b} \right){s^q} + a{s^q}{{\rm{e}}^{ - s\tau }} + \\ \;\;\;\;\;\;\;\;\;a{K_2}{{\rm{e}}^{ - s\tau }} - {K_2}\left( {b - {K_1}} \right) = 0 \end{array} $

(15)

Let

$ \left\{ {\begin{array}{*{20}{l}} {{\psi _1} = {K_1} + {K_2} - b}\\ {{\psi _2} = a}\\ {{\psi _3} = a{K_2}}\\ {{\psi _4} = - {K_2}\left( {b - {K_1}} \right)} \end{array}} \right. $

Simplify Eq. (15), then we can obtain

$ {s^{2q}} + {\psi _1}{s^q} + {\psi _2}{s^q}{{\rm{e}}^{ - s\tau }} + {\psi _3}{{\rm{e}}^{ - s\tau }} + {\psi _4} = 0 $

(16)

$s=\text{i} w=|w|\left(\cos \frac{\mathtt{π}}{2}+i \sin \left(\pm \frac{\mathtt{π}}{2}\right)\right) $ is a root of Eq. (16), when w>0, then isin $\frac{\mathtt{π}}{2} $, while when w < 0, then －isin $\frac{\mathtt{π}}{2} $, where w is real-number. Therefore, we separate the real and imaginary part to obtain

$ \left\{ \begin{array}{l} |w{|^{2q}}\cos q\mathtt{π} + {\psi _1}|w{|^q}\cos \frac{{q\mathtt{π} }}{2} + \left( {{\psi _2}|w{|^q}\cos \frac{{q\mathtt{π} }}{2} + } \right.\\ \;\;\;\;\left. {{\psi _3}} \right)\cos \tau w + {\psi _2}|w{|^q}\sin \left( { \pm \frac{{q\mathtt{π} }}{2}} \right)\sin \tau w + {\psi _4} = 0\\ |w{|^{2q}}\sin ( \pm q\mathtt{π} ) + {\psi _1}|w{|^q}\sin \left( { \pm \frac{{q\mathtt{π} }}{2}} \right) - \\ \;\;\;\;\left( {{\psi _2}|w{|^q}\cos \frac{{q\mathtt{π} }}{2} + {\psi _3}} \right)\sin \tau w + \\ \;\;\;\;{\psi _2}|w{|^q}\sin \left( { \pm \frac{{q\mathtt{π} }}{2}} \right)\cos \tau w = 0 \end{array} \right. $

(17)

Simplify Eq. (17), then

$ \left\{ {\begin{array}{*{20}{l}} {{A_1} + B\cos \tau w + C\sin \tau w = 0}\\ {{A_2} - B\sin \tau w + C\cos \tau w = 0} \end{array}} \right. $

(18)

where

$ \left\{ {\begin{array}{*{20}{l}} {{A_1} = |w{|^{2q}}\cos q\mathtt{π} + {\psi _1}|w{|^q}\cos \frac{{q\mathtt{π} }}{2} + {\psi _4}}\\ {{A_2} = |w{|^{2q}}\sin ( \pm q\mathtt{π} ) + {\psi _1}|w{|^q}\sin \left( { \pm \frac{{q\mathtt{π} }}{2}} \right)}\\ {B = {\psi _2}|w{|^q}\cos \frac{{q\mathtt{π} }}{2} + {\psi _3}}\\ {C = {\psi _2}|w{|^q}\sin \left( { \pm \frac{{q\mathtt{π} }}{2}} \right)} \end{array}} \right. $

(19)

From Eq. (18), we can get

$ {B^2} + {C^2} = A_1^2 + A_2^2 $

(20)

Then

$ \begin{array}{l} |w{|^{4q}} + {F_1}|w{|^{3q}} + {F_2}|w{|^{2q}} + {F_3}|w{|^q} + \\ \;\;\;\;\;{F_4} = 0 \end{array} $

(21)

where

$ \left\{ {\begin{array}{*{20}{l}} {{F_1} = 2{\psi _1}\left( {\cos q\mathtt{π} \cos \frac{{q\mathtt{π} }}{2} + \sin ( \pm q\mathtt{π} )\sin \left( { \pm \frac{{q\mathtt{π} }}{2}} \right)} \right)}\\ {{F_2} = 2{\psi _4}\cos q\mathtt{π} + \psi _1^2 - \psi _2^2}\\ {{F_3} = 2\cos \frac{{q\mathtt{π} }}{2}\left( {{\psi _1}{\psi _4} - {\psi _2}{\psi _3}} \right)}\\ {{F_4} = \psi _4^2 - \psi _3^2} \end{array}} \right. $

(22)

Apparently, for 0 < q < 1, Eq. (21) has no real solution if and only if F_i>0(i=1, 2, 3, 4) and ψ₃+ψ₄≠0, meaning that Eq. (15) has no purely imaginary root for all τ≥0. Thus we need to satisfy the following condition

$ {K_1} \ge b,{K_2} \ge 0 $

(23)

Suppose that Eq. (15) has two negative eigenvalues, which means if all the roots have negative parts for Eq. (15) with τ=0, all the principal minors Δ_i(i=1, 2) must be non-negative.

$ {\mathit{\Delta} _1} = {\psi _1} + {\psi _2} > 0,{\mathit{\Delta} _2} = \left| {\begin{array}{*{20}{l}} {{\psi _1} + {\psi _2}}&0\\ 1&{{\psi _3} + {\psi _4}} \end{array}} \right| > 0 $

Thus we can obtain the following condition holds

$ {K_1} > b - a,{K_2} > 0 $

(24)

On the basis of Corollary 3^[39], we drew the following conclusions:

Stability criterion of Eq.(15): System (15) with zero solution is Lyapunov globally asymptotically stable if and only if K₁ and K₂ meet both conditions (23) and (24), which means that system (8) is synchronized with system (10).

5 Simulations Research

The parameters b, c, and d in system (8) are the same as those in Section 3. The suitable control parameters satisfying conditions (23) and (24) were selected as K₁=K₂=5. The initial values of system (8) and (10) were set as (x₁(0), x₂(0))=(－4, 5) and (y₁(0), y₂(0))=(8, －4), so the initial condition of the error Eq. (14) was (e₁(0), e₂(0))=(12, －9). Let a= 1.795, q=0.84, τ=0.9, as shown in Fig. 4, then system (8) synchronized with system (10). Meanwhile, the error synchronization was zero in Fig. 5.

5.1 Effect of Parameter a

In order to stabilize the chaotic state of the system (8) as the parameter a varies, the effects of the parameter a on synchronization were discussed when setting q=0.84, τ=0.9, a=1.795, 1.9, 2.2, as shown in Fig. 6. It is clear that the synchronization state of e₁ was slower as the parameter a increased in Fig. 6(a). Since the error e₂ of system (14) was independent of the parameter a, the synchronization state of e₂ remained unchanged with the increase of a, as shown in Fig. 6(b).

5.2 Effect of Fractional Order q

When fixing a=1.795, τ=0.9, simulations about the effects of the order q on synchronization are shown in Figs. 7 by giving three cases of the order q, namely, 0.75, 0.84, and 0.98. It can be found that the faster the order increased, the slower the error e₁ trended to synchronize, and the faster the error e₂ tended to synchronize.

5.3 The effect of time delay τ

In this part, we discuss the impact of time delay τ on the synchronization. For a=1.795, q=0.84, three cases of the time delay τ, i.e., 0.7, 0.8, and 0.9 were analyzed and the simulation outcomes of the effects of the time delay τ on synchronization are depicted in Figs. 8. As shown in Fig. 8(a), the synchronization error e₁ went faster with the decrease of time delay τ. The e₂ of the error system (14) was unrelated to the time delay τ, and thus the synchronization of the e₂ remained unchanged with the increase of τ in Fig. 8(b).

6 Conclusions

This paper mainly introduces a fractional-order time-delayed memristive differential system without equilibrium point. As the parameter a changed, complex and rich chaotic attractors emerged in the system. Sufficient stability criteria of the factional-order delayed system were derived to deal with chaos synchronization of the proposed systems. Besides, the parameter a, fractional order q, and time delay τ were used as bifurcation parameters to study the impact of the system on synchronization. Results showed that as the parameter a increased, the synchronization state of e₁ went slower and the synchronization of e₂ remained unchanged. With the growth of the order q, the slower the error e₁ trended to synchronize, and the faster the error e₂ tended to synchronize. The synchronization of the error e₁ was faster with the decrease of the time delay τ, while the e₂ of the error system (14) was independent of the time delay τ, and thus the synchronization of the e₂ remained unchanged with the increase of τ. The simulation results proved that the parameter a, order q, and delay τ have certain impacts on chaos synchronization, which verified the effectiveness and feasibility of the proposed method.