Memristor-based Chaotic System with Upper-Lower Chaotic Attractors and Abundant Dynamical Behaviors


 Memristor, a controllable nonlinear element, is able to produce the chaotic signal easily. Most of the current researchers concentrate on the nonlinear properties of memristor, whereas its ability to control and adjust chaotic systems is often neglected. Thus a memristor-based chaotic system is designed to generate double-scroll chaotic attractors in this paper. The key features of the system are as follows: (a) when the polarity of the strength of memristor is adjusted, the upper-lower double-scroll chaotic attractors will be presented in the system. Chaotic motion of the system will be weakened and suppressed by properly selecting the strength of memristor, which enables the system to generate chaotic signals or suppress chaotic interference; (b) The system has very abundant dynamical behaviors, including sustained chaos, bistability, coexisting attractors, transient chaos, transient period and intermittency. To further explain the complex dynamics of the system, several basic dynamical behaviors, such as dissipation, symmetry, the stability of the equilibria, Poincare-maps, offset boosting control, recurrence analysis, 0-1 test analysis, and instantaneous phase analysis are displayed, either analytically or numerically. Moreover, the analogy circuit of the system is constructed. Experiment results prove that the PSPICE simulation and numerical analysis are consistent, which verifies the chaotic system's capability that produces chaos. Additionally, the proposed chaotic system has a strong immunity to any Gaussian noise with the zero mean. Therefore, a convenient method is employed to detect a weak multi-frequency signal embedded in the Gaussian noise based on the proposed chaotic system and the recursive back-stepping controller. The method based on the memristor-based chaos system provides a new train of thought for detecting weak signals, which is of great significance to promote the application of memristor.


Introduction
Since the memristor's existence was predicted and mathematically proved by Chua in 1971 [1], it is considered the fourth fundamental circuit element except for the resistor, the capacitor, and the inductor.Since it is difficult to realize the memristor physically, almost all the research about memristor has been carried out only in theory.The Williams team of HP Labs made the world's first Pt / TiO2 / Pt physical device with three memristive fingerprints until 2008 [2].After that, memristor has attracted researchers more attention all over the world.The memristor is expected to have a wide range of applications thanks to its memory properties, such as information storage [3], neural networks [4], and brain-like computers [5].
Moreover, due to the memristor's nonlinear properties, the memristor-based chaotic circuits are easy to generate chaotic signals.Thus many researchers focus on the research on the memristor-based chaotic system [6][7][8][9][10][11][12].Itoh and Chua [13] proposed the memristor-based chaotic circuit by replacing Chua's diode in Chua's circuit with a flux-controlled piecewise linear memristor model, which is the first memristor-based system.As a result of their works, the growing people's focus on chaotic systems has reached a noteworthy milestone.In 2017, Kengne et al. [14] proposed a memristorbased system derived from the autonomous jerk circuit [15] using a first-order memristor diode bridge to replace the semiconductor diode of the original circuit.In 2018, Wang et al. [16] designed and analyzed a new memristor-based chaotic system that shows a coexisting multi-stability.In 2020, Zhang et al. [17] constructed a new memristor-based Hopfield neural network (HNN) model with multi-double-scroll attractors, showing its coexisting characteristics.The same year, Zhang and his colleagues [18] applied a flux-controlled memristor to modify a variable-bootable chaotic system [19] such that it exhibits relatively robust chaotic oscillation compared with other multi-stable memristor-based systems.In the above literature, most scholars follow with interest in the natural nonlinearity of memristor while ignoring the controllability of its resistance variability.Meanwhile, the research of adjustable characteristics of the memristor in memristor-based circuits is still relatively less.Thus the goal of this paper is to establish a correlation between the strength of memristor and the chaotic systems.
Next, chaos theory is one of the best choices in weak signal detection because of the high sensitivity of the chaotic systems to initial conditions and robust noise immunity.This kind of method is entirely different from the traditional methods of weak signal detection, which have many limitations, such as high signal-to-noise ratio (SNR) requirements, limited detection accuracy, signal, and noise would be amplified together, possible signal loss due to filtering, and so on [20].Due to its unique advantages, there is much research on signal detection based on the chaotic system.In 1992, Birx and Pipenberg [21] combined the chaotic oscillator with a complex map feedforward neural network to detect weak signals submerged in Gaussian noise for the first time.Xu et al. [22] used both a chaos system and proportional differential control to detect an unknown signal's frequency.Li and Zhang [23] presented a method for weak sinusoidal signals embedded in strong noise by establishing two chaotic systems.Luo et al. [24] presented a memristor-based chaotic system on the basis of the classical Chua system and implemented weak signal detection applications.Thus, in the memristor-based chaotic system proposed in this paper, a recursive back-stepping controller is utilized to detect a multi-frequency weak signal.It is of vital significance to realize the application and development of memristor.
Additionally, the goal of this paper is to design a memristor-based chaotic system, which can exhibit more complex dynamics.For this purpose, a memristor-based chaotic system, which is a simple circuit structure and possesses complex dynamical behaviors, such as sustained chaos, bistability, coexisting attractors, transient chaos, transient period and intermittency.Moreover, to further explore the applications of the system, a convenient method is employed to detect a weak multi-frequency signal embedded in the Gaussian noise based on the memristor-based chaotic system and the recursive back-stepping controller.
The rest of the paper is organized as follows.In Section 2, the novel four-dimensional memristor-based chaotic system produces a two-scroll attractor is presented.Moreover, the influence of the strength of the memristor on the system is analyzed in detail, and the nonlinear behavior of the system is investigated.In Section 3, the essential dynamic characteristics of the system are proved.In Section 4, an electronic circuit of the chaotic system is established, and circuit experiments validate the theoretical analyses and MATLAB simulations.In Section 5, a detection method of weak signal frequency based on the memristor-based chaotic system is carried out.The concluding remarks are summarized in Section 6.

mathematical model
Lü proposed a 3-D autonomous system, displaying coexistence attractors with upper and lower symmetry in 2004 [25].Thus a 4-D memristor-based chaotic system based on the Lü system is established.Its state equation can be described as follows: where and y is the input flux of the memristor.M(0) denotes the initial value of memristor.Ron and Roff are limiting values of the memristor resistance, and where uv and D represent the average mobility of oxygen deficiencies and the thicknesses of film, respectively.In this paper, the parameters of memristor model are list as follows: Roff = 20 kΩ, Ron = 100 Ω, M(0)=16 kΩ, D = 10 nm，uv = 10 -14 m 2 s -1 v - 1 .

The influence of the strength of memristor on the system
An intriguing dynamic phenomenon of upper-lower chaotic attractors will be present in the system (1).Suppose that the parameter a = 2, b = 8.2, c=5, d = 3, g = 5 with initial condition (-0.1, -0.1, -0.1, -0.1), the strength of memristor r =10000 and r=-10000 are selected, respectively.Then system (1) can generate completely separated upper-lower double-scroll chaotic attractors, shown in Figs.2(a)-(c).The upper one is a doublescroll chaotic attractor with the strength of memristor r =-10000(red), and the lower is also two-scroll chaotic attractors with the strength of memristor r =10000(blue).The time-domain waveform of state variable w is shown in Fig. 2(d).It can be concluded from plots that the system (1) can generate upper-lower two-scroll attractors, and the attractors of positive and negative polarity systems are located in the negative and positive domain in the phase space, respectively.Therefore, we only need to analyze the influence of r on the system (1) when r is positive, and the influence of r on the system (1) can be known in the whole real range.Meanwhile, when 53000 r = , the system (1) transforms from the chaotic state of double-scroll with upper-lower to the chaotic state of single-scroll with upper-lower is shown in Fig. 3 (a), where the Lyapunov exponents are L1 = 0.3406, L2 = 0, L3 = -5.002and L4=-12.2086when r=53000.When 72000 r = , the Lyapunov exponents are L1 = 0.0051, L2 = -0.4123,L3 = -3.9649and L4=-12.8657for r = 72000.The phase trajectory of the y-w plane is depicted in Fig. 3(b), where the system (1) performs the upper -lower limit cycle motion.When 720000 r = , the phase trajectory of the y-w plane is shown in Fig. 3 (d), and the Lyapunov exponents of the system are calculated as L1 = 0.0017, L2 = -0.2837,L3 = -5.2884and L4=-8.2723 for r = 720000.It can be seen that system (1) is in a periodic state, and the chaotic motion is not generated.Moreover, it's obvious that the upper periodic attractor and the lower periodic attractor are completely independent.By comparing the maximum Lyapunov exponent when r takes different values, they are demonstrated that the chaotic state of system (1) is gradually weakened.Generally, by choosing r appropriately, the corresponding attractors will be produced when the polarity of r is changed with the same initial value.Also, by increasing the value of r, the chaotic motion of the system (1) is no longer generated.Therefore, chaos and the periodic phenomenon will be displayed by choosing different r values, and the application range of the system (1) will be expanded through flexibly selecting r.The system (1) can be applied in secure communication and other practical applications that need to produce chaotic motion and can be applied in the process of suppressing chaos due to its ability to suppress chaos.Consequently, a bridge between the strength of memristor and chaotic systems is built.Moreover, such a chaotic system, which could generate double-scroll chaotic attractor, single-scroll chaotic attractor, and periodic attractor for just simple parameter change, can be very significant in chaotic secure communication systems.

Impacts of initial conditions
To analyze the sensitivity of the initial condition, the parameters of system (1) are determined as a = 2, b = 8.2, c=5, d = 3, g = 5 and r=10000 and the initial conditions are considered as (0.1, 0.1, 0.1, w (0)).Fig. 4 shows the bifurcation diagram and Lyapunov exponent spectrum with respect to initial value w (0), and other initial values are configured as x (0) = y (0) = z (0) =0.1.The phenomenon of sustained chaos is observed, and its maximum Lyapunov exponent is always positive and can be considered as a constant in the interval of.Simultaneously, the bifurcation diagram also shows the sustained chaotic state of the system (1).However, the memristor-based chaotic system's dynamical behavior is quite different for the system parameter a = 1.6, and other parameters remain unchanged.When z (0) and w (0) are adjustable in [-1, 1] and [-2, 2] regions respectively, the dynamic behaviors, which are characterized by bifurcation diagram and the first two Lyapunov exponents and depend on the initial conditions, obtained by numerical simulation are shown in Figs.5(a)-5(b).They can be observed that the long-term dynamic behavior composed of unstable chaos and period is related to the initial conditions, which leads to the generation of bistability of the memristor-based chaotic system.Furthermore, the Lyapunov exponent of Fig. 5(a) is invariant in the unstable chaotic region, which indicates the robustness of chaotic behavior.Taking Fig. 5(a) as an example, system (1) starts from the chaotic state.Then the system (1) jumps from the chaotic state into the periodic state at z (0) = -0.87.Next, the periodic state is transformed into chaotic state by tangent bifurcation at z (0) =-0.15.There exist some periodic windows in the range of -0.15 < z (0) ≤ 0.36.Next, the system (1) switches to the periodic state as z (0) = 0.36.Especially, the period-2 attractor in the z-w plane is exhibited in Fig. 5(e) for z (0) = -0.5, whereas Fig. 5(f) proves chaotic state of the system (1) for z (0) = 0.1.In order to confirm the bistability behavior of the memristor-based chaotic system, the basins of attraction in the z (0)-w (0), x (0)-z (0) and y (0)-z (0) planes are illustrated in Fig. 6.In Fig. 6 (a), x (0) = y (0) = 0.1, while in Fig. 6 (b) and Fig. 6 (c), y (0) = w (0) = 0.1 and x (0) = w (0) = 0.1, and the yellow and blue regions represent chaotic attractors and periodic attractors, respectively.It is obvious from Fig. 6 that the initial values play an important role in the dynamic behavior of the system (1).In addition, it should be emphasized that the yellow region of instability shown in Fig. 6 is closely related to other initial conditions x (0), y (0) and w (0).

The influence of parameter
To explore the system's bifurcation behavior (1), the system parameter c is taken as the control parameter and varies in the interval [0, 20].The Lyapunov exponent spectrum and bifurcation diagram of the system (1) varying with parameter c are shown in Fig. 7 (a) and Fig. 7 (b), respectively.It can be seen from Fig. 7 (a) that the maximum Lyapunov exponent varying with parameter c in [3.09, 15.2] intervals are greater than zero, indicating that chaos can be generated in the interval.As can be seen from Fig. 7 (b), it is evident that the system (1) is in the chaotic state in the interval.The stable and unstable regions of the system reflected by the trajectories in Fig. 7(a) and Fig. 7(b) are consistent, and there are several prominent periodic windows in fig.7(a).
With the change of system parameter c, the phase orbit diagram of chaotic attractor and periodic orbit in the y-w plane is shown in Fig. 8. Figs.8 (a)-(c) reflect the system's process from period-doubling bifurcation to the chaotic state, in which the attractors of period-1, period-2, and single-scroll are generated.At c = 3.31, however, the trajectory has a sudden change from top to bottom.A double-scroll chaotic attractor is generated in the interval [4, 15.2] system, corresponding to fig.8(d).Until a > 15.2, the antiperiod doubling bifurcation of the system exits chaos and reaches the steady state.The corresponding phase diagrams of this process are shown in Fig. 8 (e)-(f).Fig. 8 intuitively reflects the whole process of the system from period-doubling bifurcation to chaos, and finally to exit chaos through anti-period doubling bifurcation.
In order to further explore the influence of two parameters on the system (1) concurrently, the dynamic diagrams of the system (1) with respect to parameters a and b, c and d are depicted in Fig. 9 (a) and Fig. 9 (b) respectively, where the initial conditions and other parameters are the same as Section 2.1.In the dynamical maps, when the parameters are in the red region, the system (1) is in a periodic state, and the system is in the chaotic state when the value in the blue, green, yellow, and purple region.It should be noted that multi-parameter dynamical maps are widely applied in engineering because they provide global information about the dynamic behavior of the

Coexistence of attractors
In this section, two sets of special initial conditions (0.1, 0.1, 0.1, 0.1) and (0.1, 1, -1, 0.1) are discussed, and the orbit from initial condition (0.1, 0.1, 0.1, 0.1) is marked in blue, while the orbit from initial condition (0.1, 1, -1, 0.1) is marked in red.When parameter a changes in the range of 0 to 5, the maximum Lyapunov exponent and bifurcation diagram of coexistence are shown in Figs.10(a)-(b).Comparing with Fig. 10 (a) and Fig. 10 (b), it is not difficult to observe that the stable state region and unstable state region of the system reflected by the orbit are consistent with each other.The coexisting double-scroll chaotic attractors, coexisting single-scroll chaotic attractors, and coexisting limit cycles with the different number of periods and coexisting point attractors can be observed.
When a < 1.38, the maximum Lyapunov exponent is equal to 0, and the system (1) shows coexisting period-1 limit cycles.In the region of 1.38 < a < 1.49, the perioddoubling bifurcation results in the coexisting period-2 limit cycle.With the growth of parameter a, the system (1) behaves as a coexisting period-4 limit cycle when the parameter a satisfies the interval [1.49, 1.52].When the interval is 1.52 < a < 1.66, the system (1) produces a coexisting single-scroll chaotic attractor with weak chaotic behavior.At a =1.66, the two coexisting orbits have state mutation from top to bottom or from bottom to top.In the range of [1.66, 1.76], a coexisting single-scroll attractor is produced again.Then the system enters into the coexisting double-scroll attractor is located in [1.77, 3.94].After parameter a > 3.94, the dynamics of the system changes from chaotic behavior to point attractor behavior with the further evolution of parameter a, and is stable on the point attractor.
For different values of a, the phase orbits of the system on the z-w plane can be drawn by numerical calculation, as shown in Fig. 11 (a) -(f), respectively.The orbit marked in blue corresponds to the initial condition of (0.1, 0.1, 0.1, 0.1), while the orbit marked in red corresponds to the initial value of (0.1, 1, -1, 0.1).Typical coexisting attractors are shown in Table 1.The existence of coexisting attractors by varying the initial state also fully demonstrates the chaotic system's extreme sensitivity to the initial value.For different values of a, the phase orbits of the system on the z-w plane can be drawn by numerical calculation, as shown in Fig. 11 (a) -(f), respectively.The orbit marked in blue corresponds to the initial condition of (0.1, 0.1, 0.1, 0.1), while the orbit marked in red corresponds to the initial value of (0.1, 1, -1, 0.1).Typical coexisting attractors are shown in Table 1.The existence of coexisting attractors by varying the initial state also fully demonstrate the extreme sensitivity of the chaotic system to the initial value.

Transient phenomenon and intermittency
Due to the appearance of the non-attractive saddle point in the phase space, the orbit behaves as chaotic behavior in a finite time interval until it falls into the final nonchaotic state.The well-known phenomenon is called transient chaos [27][28][29], which often occurs in nonlinear dynamics.The reason is that the boundary distance between the chaotic attractor and the basin of attraction in the phase space gradually decreases with the change of the system's parameter, and it does not meet until it reaches the critical value.At this moment, the chaotic attractor touches an unstable periodic orbit, which leads to the boundary crisis.In the region, it is easy to observe the phenomenon of transient chaos in the system (1).For example, considering the parameter a = 2, b = 8.2, c = 5, d = 5.7, g = 5, r= 10000 , and setting the initial condition as (2, -2, -2, 0.1), the system(1) produces transient chaos, that is, the trajectory of the system(1), with the evolution of time, shifts from transient chaos to periodic behavior.Fig. 12(a) shows the time domain waveform of variable z, in which the chaotic behavior is located in the time interval of [0s, 125s], and then gradually switches to periodic behavior when t > 125s.The waveform of the variable z in the time interval [0, 50s] is depicted in Fig. 12 In contrast to the transient chaos, the transient period [30] is defined as the period movement stops after a long period of time.From then on, the orbit of the system presents chaotic motion.Therefore, the unconventional transfer behavior of long-term transient period and steady-state chaos in the system is a new dynamic phenomenon, which is related to the initial conditions of state variables and completely different from the phenomenon of transient chaos and steady-state period.Configuring the parameters as a = 2, b = 10, c = 4.81, d = 3, g = 5 and r = 10000, and setting initial value as (0.1, 0.1, 0.1, 0.1), the time domain waveform of the variable z is displayed in Fig. 13(a), which indicates that the proposed system starts from periodic oscillation and then changes into chaos along with time evolution.For further explanation, the waveform of the variable z in the time interval [0, 50s] is depicted in Fig. 13(b).By applying the phase diagrams, Figs.13(c)-(d 14(a) demonstrates the intermittence on the route from chaos to periodicity.Simultaneously, the proposed system will emerge different intermittences at different initial values.Different intervals from the former are shown in Fig. 14(b), where the initial conditions are (0.1, 0.1, 0.1, -0.1).A large number of numerical simulations exhibit that this phenomenon not only depends on the system parameters, but also is extremely sensitive to the initial conditions of the system.

Symmetry, dissipativity and the existence of attractors
The system ( 1) is symmetric about the y-axis and w-axis, a simple proof is described as: ( In other words, system ( 1) is symmetrical about the two coordinate axes y and w, respectively.In addition, the symmetry persist for all parameter values of the system (1).Obviously, the coordinate axes y and w themselves are solution orbits of the system (1).Moreover, the orbit on the y-axis and w-axis towards infinity, as t → ∞.The chaotic system has a highly symmetrical structure and is robust to various small perturbations.
The divergence of the system (1) can be obtained from the vector field as follows: ( ) According to equation (6), the system (1) is dissipative.All the trajectories of the system will eventually shrink to a set of zero volume, as t→∞, and the extreme motion will converge to an attractor, thus proving existence of the attractors of system(1).

Equilibrium points and stability
In order to obtain the equilibrium points of the system, let x = y = z = w =0.The equations of system (1) are rewritten as follows: By calculation, we get that the two equilibrium points of system (1) are s1= (0, 0, 0, 0) and s2 = (10.67595,0.80757, 2.87387, -28.85548) for parameter a = 2, b = 8.2, c=5, d = 3, g = 5 and r=10000.By linearizing the system (1), the Jacobin matrix at the s1 is 1 00 0 0 0 13 0 1 0 0 , 0 0 0 0 1 5 00 00 (0) 8 and its characteristic equation is given to: Then, the eigenvalues of J1 for typical parameters above mentioned are calculated as: which shows that s1 is an unstable saddle point.Similarly, the Jacobin matrix of the system (1) at the s2 is and its characteristic equation is： Where,  Then, it is easy to calculate the eigenvalues of J2 as follow: Thus, the equilibrium point s2 is also an unstable saddle-point.

Poincare projection
As an effective way of judging whether system has chaotic behavior, Poincare projection is often used to analyze chaotic system.In this section, we take projections x=5 and y=0, respectively.The coexisting Poincare mapping on two projections of system (1) are shown in Fig. 15, in which red represents the strength of memristor r =-10000, blue represents the strength of memristor r =10000.At the same time, it is observed that the chaotic attractors of symmetric double-scroll are visualized, which further proves the characteristics of attractors shape.Moreover, it is clear that some sheets are folded.Therefore, the memristor-based double-scroll chaotic system has more complex dynamical behaviors than the general chaotic systems, and it can be easily manipulated by the strength of memristor r.In addition, the characteristic are very meaningful and worthy of further study in image encryption and chaotic secure communication.

Recurrence analysis and 0-1 test analysis
Recurrence plot (RP) [31] is an important method to analyze the periodicity, chaos and non-stationary of time series.It can reveal the internal structure of time series.Recursive plot is especially suitable for short time series data, which can test the stability and internal similarity of time series.At a different time j, the recurrence of a state at time i is marked within a two-dimensional square matrix and its row and column correspond to a pair of timescales.The two-dimensional square is marked with the white and black dots (recursion is usually represented by a black dot).By applying a time delay ( )  , signal x 1 (t), which comes from the multiple variables ( ) [ ( ), ( ), ( ), , ( )] , is used to create a lower n-D phase space.In n-D phase space, the reconstructed trajectory can be expressed as follows [ , ,..., ] − and the embedding dimension is DE.
Any recurrence of state i with state j can be represented as follows , () In equation ( 16), the Heaviside function denotes  and ε stands for an arbitrary threshold.The recurrence plot of y signal of system ( 1) is plotting for ε = 2, DE = 7 and the time delay τ = 55 by using a total of 5000 data points as shown in Fig. 16 (a).The random distribution of the black dots shows the chaotic characteristics.On the other hand, the dynamic behavior of the system (1) in the p-s plane was depicted using the 0-1 test algorithm as shown in Fig. 16(b).The phase diagram exhibits system ( 1) is an unbounded trajectory similar to Brownian motion behavior, which indicates that the system (1) has chaotic characteristics.

Instantaneous phase
Concurrently, in order to verify the chaotic characteristics of system (1), the instantaneous phase are given.The instantaneous phase is plotted by applying Hilbert transformation.Assuming that the a chaotic signal x (t) is expressed in the form of complex signal s (t), so its amplitude and phase can be obtained as follows Where P •V is the principal value of Cauchy in Hilbert transform.Hilbert transform is obtained in MATLAB by the algorithm given [28].After truncating initial transient part, the instantaneous phase of x (t) and y (t) signals are exhibited in Figs.17 Obviously, the instantaneous phase of a chaotic signal increases monotonously with time [31] , and the monotonous increase in the phase diagram demonstrates the chaotic characteristic of system (1).
Fig. 17.The instantaneous phase of the x and y signals of system (1) with x (0) = (0.1, 0. 1, 0. 1, 0.1) and t = 0.001 using the Hilbert transform.(a) the instantaneous phase of the x signal of system (1); (b) The instantaneous phase of the y signal of system (1).

Offset boosting control
The memristor-based chaotic system can also be conveniently and simply controlled by state variables w, replaced by w + u, where u is regarded as an offset variable.Consequently, system (1) has a specific dynamic phenomena after offset boosting control by means of using this replacement [32] .This offset variable u has an influence on changing and controlling phase space from bipolar signal to unipolar signal, that is, the attractors which are containing the phase space is either negative unipolar or positive unipolar being depended on the value of offset variable u.Firstly, system (1) can be rewritten as system (19) since the state variable w appears once in the first equation and the seconded equation of system (1).
and u is the offset boosting controller with ranging from -0.5 to 0.5.The parameters and initial values of system (19) are taken in accordance with those of system (1).Figs.18 (a)-(c) show that when the offset boosting controller u changes, the w is boosted from a bipolar with chaos to a unipolar one with chaos, which is illustrated by the green and purple chaotic attractors.In order to further prove this phenomenon, the waveform of state w(t) is depicted in Fig. 18(d).According to above-mentioned analysis, it is demonstrated that the memristor-based two-scroll chaotic system is are appropriate for applying in the choice of offset boosting control.Meanwhile, Fig. 18 also displays that there are abundant and complex dynamic behaviors in the system (1) due to the introduction of memristor.

Analog implementation and PSPICE simulation
This section describes the design and construction of an electronic circuit that realizes system (1), which is based on the software platform PSPICE.The combination of operational amplifiers and resistors or capacitors are used to realize the integral operation and addition and subtraction operation.Therefore, the electronic circuit of system (1) can be implemented by an electronic circuit consisted of operational amplifiers, diodes, memristor, linear resistor and capacitors.The analog circuit contains four modules, each of which corresponds to a dimensionless equation of a variable in the system (1).The detailed analog circuit implementation is shown in Fig. 19.In the circuit design, operational amplifiers LM675 are adopted and their supply voltage are taken as VEE=-30V and VCC=+30V.The voltages at the nodes labeled vx, vy, vz and vw As shown in Fig. 19, the operational amplifiers U12, U13 and two D120NQ045 diode models are employed to realize the absolute value circuit.The output voltage of U13 is given by 13 32 30 32 ,0 When we set R29=R30=R31=R32=1kΩ, R35 =500Ω, then voltage v U13 =|vy|.Two suitable matched resistors are also incorporate in U12 and U13 such as R48=R29//R30=500Ω and R49=R31//R32//R35=250Ω for circuit stability.The operational amplifier U14 is a reverse amplifier, i.e., v U13 =-(R34/R33) |vy|.When R33=R34=1kΩ, it becomes v U14 =-| vy |.The voltage v U14 is the flux input of the SPICE model of memristor.The charge terminal is connected with the operational amplifier U17.The memristor parameters are set to Ron=100Ω, Roff=100Ω, M (0) =16kΩ, D=100nm and uv =10 -14 m 2 s -1 v -1 .The function of f (•) can be realized by the memristor model.Furthermore, a selection switch S is applied to generate upper-lower attractors.When switch S connect with the reverse amplifier U19, the system (1) displays upper-attractors as follows Figs.20(a)-(c).Conversely, lower-attractors can be generated by connecting switch S directly with the reverse adder U21 ， which are shown in Figs.20(d)-(f).
The simulated results are shown in Fig. 20.The SPICE simulation time interval ranges from 0 to 500 seconds and the maximum step size is 0.005 seconds.The results illustrate that a very good similarity can be captured between numerical (see Figs.
5. Weak signal detection based on the memristor-based chaotic system

Design of the recursive back-stepping controller
Considering introduction of control term in system (1), the memristor-based system (1) is now written as where ui(t), i=1, 2, 3, 4 are the control inputs such that the state variable x, y, z, and w of system (1) can take desired values xd, yd, zd, and wd, respectively.The error states between the state variable and the desired values are given by [33]   , , , .
In order to design general control functions ui(t) that can control system (1) to track any trajectory f(t) that is a smooth function of time, we let Where ci, i=1, 2… 6 are the control parameters to be chosen appropriately.Substituting Eq. ( 24) into Eq.( 23) yields Substitution of Eq. ( 22) into the time derivative of Eq. ( 25) obtain the error system Letting a form of Lyapunov function is The error system (28), if V ̇<0, is asymptotic stability.Thus, we take the value of ui, i=1, 2, 3, 4 such that V ̇=-∑ k i e i 2 <0, i=x, y, z, w as follows ) .
Let c1=c3=1 and c2=c4=c5=c6=0, then the controller (29) reduces to In order to further observe the controller effect, the fourth-order Runge-Kutta solver ODE45 is applied for the solution of Eq. ( 1) and Eq.(30).Here, we keep the system(1) parameters a = 2, b = 8.2, c=5, d = 3, g = 5, r=10000 and set the initial condition (8,3,6,2).Thus, the Lyapunov exponents of system (1) are L1=0.5694,L2=0.0020≈0,L3=-4.9924 and L4=-13.3474,which ensures chaotic properties of the system.Taking smooth function f (t) =sin (2t) as example, the time-domain results of u1, u2, u3, u4 are shown in Fig. 21.As can be seen from this figure, the controller has an ability of tracking the system to a desired value (sin (2t), 0, 0, 0).When the controller is switched on at t =100 seconds, the state variables of system (22) move with time as shown in Fig. 22.The states x(t), y(t), z(t), w(t) move chaotically with time when the controller is switched off and are controlled to track the desired value (sin(2t), 0, 0 ,0) when the controller is switched on.In this section, the recursive back-stepping controller mentioned above is used to detect weak signal.Firstly, whether the chaotic system is in any chaotic state in the chaotic domain is judged.Secondly, the recursive back-stepping controller is used to control the system in the chaotic state to desired values.Finally, the frequency of the signal is detected by spectrum analysis.The detection schematic diagram is shown in Fig. 18.Three steps of detection process are described as follows: Step 1.Put the noise into the system when the system is in chaotic state and judge the immunity to noise.
Step 2. Put the controller into the system that can track the system to a desired value given.
Step 3. Analyze the output signal of the detection system (consist of controller and chaotically system).

Judging immunity
Chaotic systems are seemingly random and depend on the initial conditions sensitively.Besides this, they are immune to background noise.In the memristor-based system, letting ∆x, ∆y, ∆z, ∆w denotes respectively the small perturbations to x, y, z, w by Gaussian noise, and then the system equations are written as Fig. 24.Frequency spectrum of the output signal

Conclusion
In this paper, a novel memristor-based 4-D double-scroll chaotic system has been proposed.Theoretical analysis and numerical simulation demonstrate a series of dynamical properties of the new chaotic system, such as dissipation and the stability of the equilibrium point, Poincare-map and Lyapunov exponents and dimension, power spectrum, the offset boosting control, recurrence analysis, 0-1 test analysis and instantaneous phase analysis.Moreover, the memristor is capable to control and adjust the system and show the interesting effects such as generation of an upper-lower attractors by varying polarity of the strength of memristor in this work.In addition, coexistence of attractors, transient period, transient chaos and paroxysmal chaos are observed.Furthermore, PSPICE simulation is applied to illustrate consistency with the numerical simulation results, which demonstrates the feasibility of system.Finally, by designing a recursive back-stepping controller based on the novel two-scroll chaotic system, a convenient method is used to detect a weak multi-frequency signal embedded in the Gaussian noise.We hope that our contribution will enrich nonlinear dynamics x, y, z，and w are the state variables of system (1), a, b, c, d, g, and r are system (1) parameter.The parameter r is an arbitrary constant, expressed as the strength of memristor, and the function f(• ) represents the charge of memristor[26]

Fig. 2 .
the phase diagram and time-domain waveform of the system (1).(a) x-w plane; (b) y-w plane; (c) z-w plane; (d) time-domain waveform of w.

Fig. 10 .
coexisting bifurcation diagram and coexisting maximum Lyapunov exponent diagram with different initial conditions.(a) coexisting bifurcation diagram; (b) coexisting maximum Lyapunov exponent diagram.

Fig. 12 .
(b) for clarity.In order to further illustrate the transient chaos phenomenon, a chaotic phase diagram is shown in Fig. 12 (c) for the interval [0s, 100s], and a periodic phase diagram is displayed in Fig. 12 (d) for the interval [300s, 400s].Transient chaos.(a) time-domain waveform of the variable z in the time interval [0, 500 s];(b) time-domain waveform of the variable z in the time interval [250, 300 s]; (c) phase diagram of transient chaos; (d) phase diagram of periodic oscillation

Fig. 20 .
1(a) -(c)) and SPICE simulated results.Experimental results of memristor-based system (1) using PSPICE software.(a)the upper attractor in x-w plane; (b)the upper attractor in y-w plane; (c)the upper attractor in z-w plane; (d) the lower attractor in x-w plane; (e)the lower attractor in y-w plane; (f)the lower attractor in z-w plane;

5. 2
Detection of weak signal frequency based on the memristor-