An Integer-Order Memristive System with Two-to Four-Scroll Chaotic Attractors and Its Fractional-Order Version with a Coexisting Chaotic Attractor

First, based on a linear passive capacitor C, a linear passive inductor L, an active-charge-controlled memristor, and a fourth-degree polynomial function determined by charge, an integer-order memristive system is suggested. The proposed integer-order memristive system can generate two-scroll, three-scroll, and four-scroll chaotic attractors. The complex dynamics behaviors are investigated numerically. The Lyapunov exponent spectrum with respect to linear passive inductor L and the two-scroll, threescroll, and four-scroll chaotic attractors are yielded by numerical calculation. Second, based on the integer-order memristive chaotic system with a four-scroll attractor, a fractional-order version memristive system is suggested. The complex dynamics behaviors of its fractional-order version are studied numerically. The largest Lyapunov exponent spectrum with respect to fractional-order p is yielded. The coexisting two kinds of three-scroll chaotic attractors and the coexisting three-scroll and four-scroll chaotic attractors can be found in its fractional-order version.

Motivated by the above considerations, first, based on a memristor-based chaotic circuit reported by Muthuswamy and Chua [24], Bao et al. [25], and Teng et al. [26], an integer-order memristive chaotic system with two-scroll, three-scroll, and four-scroll chaotic attractors is provided in this paper.It is noticed that there is only a single-scroll chaotic attractor in [24], only a double-scroll chaotic attractor in [25], and only double-scroll and four-scroll chaotic attractors in [26].However, there are two-scroll, three-scroll, and four-scroll chaotic attractors in our memristive system.Meanwhile, the Lyapunov exponent spectrum, and phase diagram for our memristive chaotic system are obtained.Second, based on the proposed integer-order memristive chaotic system with a four-scroll chaotic attractor, a fractional-order version chaotic system is suggested.We find that the coexisting three-scroll and four-scroll chaotic attractors and coexisting two kinds of three-scroll chaotic attractors are emerged in the fractional-order version.To the best of our knowledge, this result is rarely reported.
The outline of this paper is organized as follows.In Section 2, the concept of a memristor and some memristorbased system are briefly reviewed.Based on the review, we present an integer-order memristive chaotic system with two-scroll, three-scroll, and four-scroll chaotic attractors and some basic dynamics behaviors are obtained.In Section 3, based on the integer-order memristive chaotic system with a four-scroll chaotic attractor, we present its fractional-order version and we find that there are coexisting chaotic attractors in its fractional-order system.In Section 4, the conclusion is given.

An Integer-Order Memristive Chaotic System
The charge-controlled memristor [24,26] is described by a nonlinear I-V characteristic as V M = M q I M and q = F q, I M .Here, V M , I M , and q are the voltage, current, and charge associated to the device, respectively.M q is the memristance, and F q, I M is the internal state function.In [24,26], two schematics of the simplest memristor-based chaotic circuit with a linear passive inductor, linear passive capacitor, and a nonlinear active memristor have been reported.The state equations represent the current-voltage relation for the linear passive capacitor, and the inductor is described as where V C denotes the voltage of the linear passive capacitor C and I L denotes the current of the linear passive inductor L.
In [24], the memristance M q is defined as M q = β q 2 − 1 , and the internal state function F q, I M is defined as F q, I M = I M − α + I M q, where I M = −I L .The memristor-based circuit in [24] has a single-scroll chaotic attractor (for more details, see [24]), and its dynamics are described by In [26], the memristance is chosen as M q = δq 4 + γq 2 − β and the internal state function is chosen as F q, I M = I M − α − I 2 M q, where I M = −I L .The memristor-based circuit in [26] has double-scroll and four-scroll chaotic attractors (for more details, see [26]), and its dynamics are shown as Now, based on [24,26], an integer-order memristive system is suggested in our paper.The memristance is defined as M q = δq 4 − β, and the internal state function is defined as So, the integer-order memristive chaotic system in this paper is suggested as where C = 1F, δ = 0 5, β = 2 4, α = 0 75, and 1H ≤ L ≤ 8H.
The equilibrium points of system (4) can be calculated by Obviously, only I L , V C , q = 0, 0, 0 is the equilibrium point in system (4).The Jacobian matrix J at this equilibrium point is and its eigenvalues are So, the equilibrium point I L , V C , q = 0, 0, 0 in system (4) is unstable.

Complexity
By numerical calculation, the Lyapunov exponent spectrum of integer-order memristive system (4) with respect to linear passive inductor L can be obtained and is displayed in Figure 1.
According to Figures 2 and 3, we find that two kinds of three-scroll chaotic attractors are emerged in our integerorder memristive chaotic system.Figure 5: Two-scroll chaotic attractor in integer-order memristive system (4).Complexity According to the above results, the proposed integerorder memristive chaotic system (4) in this paper can generate two-to four-scroll chaotic attractors.This result is different with many previous results [21,[23][24][25][26][27][28].

A Fractional-Order Memristive Chaotic System with Coexisting Chaotic Attractors
In this section, based on integer-order memristive chaotic system (4), a fractional-order version with coexisting chaotic attractors is given.
According to Figure 4 in Section 2, the four-scroll chaotic attractor is emerged in integer-order memristive system (4) with C = 1F, δ = 0 5, β = 2 4, α = 0 75, and L = 1 4. Now, based on this case, a fractional-order version memristive system is suggested, which is shown as follows: Here, 0 92 ≤ p ≤ 1 is the fractional-order version and Now, by the improved version of Adams-Bashforth-Moulton numerical algorithm [36], nonlinear fractionalorder system (7) with initial condition (I L 0 , V C 0 , q 0 ) can be discretized as follows: α j,n+1 I L j , where The approximation error is as follows: In this numerical algorithm, T is the total time length of numerical calculation, N is the iterative calculation time, and τ = T/N is the step length.So, t n = nτ n = 0, 1, 2, … , N .
Next, we study the dynamical behaviors for fractionalorder system (7) by the improved version of Adams-Bashforth-Moulton numerical algorithm [36].First, using numerical calculation, the largest Lyapunov exponents (Largest LE) of fractional-order system (7) with respect to fractional-order p can be obtained, which is shown in Figure 6.
According to Figure 6, the largest Lyapunov exponent is larger than zero for 0 92 ≤ p ≤ 1.The positive largest Lyapunov exponent indicates that the chaotic attractor is emerged in fractional-order system (7).Next, some results are shown as follows: 3.1.Coexisting Three-and Four-Scroll Chaotic Attractors in System (7) for p = 0 935.Letting p = 0 935, the Largest LE is 0.3251.Therefore, fractional-order system (7) has chaotic behavior.The chaotic attractor can be obtained by numerical calculation.Here, we find that there are coexisting threescroll and four-scroll chaotic attractors which depend on the initial conditions.For example, let the initial condition be (−2,−1,−1) and (−2,1,1).The four-scroll chaotic attractor (black line) and three-scroll chaotic attractor (red line) are shown in Figure 7.

Coexisting Two Kinds of Three-Scroll Chaotic Attractors
in System (7) for p = 0 94.Letting p = 0 94, the Largest LE is 0.3864.Therefore, fractional-order system (7) has chaotic behavior.The chaotic attractor can be obtained by numerical calculation.Here, we find that there are coexisting two kinds of three-scroll chaotic attractors which depend on the initial conditions.For example, let the initial condition be (−2,−1,−1) and (−2,1,1).The two kinds of three-scroll chaotic attractors (black line, red line) are shown in Figure 8. (7) for p=0.99.Letting p = 0 99, the Largest LE is 0.2247.Therefore, fractionalorder system (7) has chaotic behavior.By numerical calculation, we find that the four-scroll chaotic attractor is emerged in fractional-order system (7).The four-scroll chaotic attractor is shown in Figure 9.

Four-Scroll Chaotic Attractor in System
According to Figure 9, four-scroll chaotic attractor is emerged in fractional-order system (7).This result is just as that of integer-order memristive chaotic system (4) with L = 1 4.
According to Figure 8, the coexisting two kinds of threescroll chaotic attractors are obtained in fractional-order system (7) and the two kinds of three-scroll chaotic attractors do not exist in integer-order memristive chaotic system (4) with L = 1 4. So, two kinds of three-scroll chaotic attractor are newly produced.
According to Figure 7, the coexisting three-scroll and four-scroll chaotic attractors are emerged in fractional-order system (7).But, there is only a four-scroll chaotic attractor in integer-order memristive chaotic system (4) with L = 1 4. So, the three-scroll chaotic attractor is newly produced.
In summary, for integer-order memristive chaotic system (4) with L = 1 4, there is only a four-scroll chaotic attractor.However, for its fractional-order version, it can produce two kinds of new three-scroll chaotic attractors and has coexisting three-scroll and four-scroll chaotic attractors.These results in Section 3 are rarely reported in the previous literature.

Conclusions
By a linear passive capacitor C, a linear passive inductor L, and an active-charge-controlled memristor, an integer-order    5 Complexity memristive system is devised in this paper.The memristance M q is defined as a fourth-degree polynomial function determined by charge, that is, M q = δq 4 − β.By numerical calculation, the Lyapunov exponent spectrum of the proposed memristor-based chaotic circuit with respect to linear passive inductor L is yielded.The proposed integer-order memristive system can generate two-scroll, three-scroll, and four-scroll chaotic attractors for suitable linear passive inductor L.
Furthermore, based on the proposed integer-order memristive system with a four-scroll chaotic attractor for L = 1 4, a fractional-order version memristive system is given.By numerical calculation, we obtain the largest Lyapunov exponent with respect to fractional-order p.This fractional-order version memristive system can newly produce two kinds of three-scroll chaotic attractors, and the coexisting threescroll and four-scroll chaotic attractors are obtained.

Figure 6 :
Figure 6: The Largest LE varies as fractional-order p.

Figure 8 :
Figure 8: The coexisting two kinds of three-scroll chaotic attractors in system (7).