Tri-valued Memristor-based Hyper-chaotic System with Hidden and Coexistent Attractors

Recently, the nonlinear dynamics of memristor has attracted much attention. In this paper, a novel four-dimensional hyper-chaotic system (4D-HCS) is proposed by introducing a tri-valued memristor to the famous L¨u system. Theoretical analysis shows that the 4D-HCS has complex chaotic dynamics such as hidden attritors and coexistent at-tractors, and it has larger maximum Lyapunov exponent and chaotic parameter space than the original L¨u system. We also experimentally analyze the dynamics behaviors of the 4D-HCS in aspects of the phase diagram, Poincar´e mapping, bifurcation diagram, Lyapunov exponential spectrum, and the correlation coefﬁcient, and the analysis results show the complex dynamic characteristics of the proposed 4D-HCS. In addition, the comparison with binary-valued memristor-based chaotic system shows that the 4D-HCS has unique characteristics such as hyper-chaos and coexistent attractors. To show the easy implementation of the 4D-HCS, we implement the 4D-HCS in an analogue circuit-based hardware platform, and the implementation results are consistent with the theoretical analysis. Finally, using the 4D-HCS, we de-sign a pseudorandom number generator to explore its potential application in cryptography.


Introduction
Chaos is a pseudorandom phenomenon produced by a certain nonlinear system.It shows many unique properties such as initial sensitivity and ergodicity.Since the first chaotic system was designed by Lorenz in 1963 [1], researchers have developed many chaotic systems and applied them to a wide body of research fields such as dynamics research [2], neural networks [3], secure communication [4][5][6], image encryption [7][8][9][10][11].
Generally, the complexity of a chaotic system is determined by its nonlinear term [12].Memristor is a circuit element with nonlinear characteristics, and it can be used to construct chaotic circuits [13].In 2008, Itoh and Chua first introduced the memristor into chaotic system to propose a memristive chaotic system [14].In 2012, Wang et al. first proposed a memristor model using light dependent resistor (LDR) [15], and then designed a chaotic circuit based on the LDR memristor [16].
In recent years, chaotic systems are discovered to have many new characteristics such as hidden attractors, coexisting attractors, and multistability [17].For example, Li et al.
proposed a new memristive chaotic system with an infinite equilibrium plane in [18].The amplitude and frequency of the system can be changed by adjusting the initial value of the internal state variable u of the memristor, indicating that the chaotic system has hidden and coexisting attractors.In 2018, using the Wien-bridge chaotic circuit, Ye et al. proposed a memristive hyper-chaotic circuit with coexisting attractors [19].In the same year, Tan et al. proposed an inductor-free memristive chaotic circuit with three line equilibria and coexisting attractors [20].In 2019, Wang et al. proposed a chaotic oscillator using memcapacitor and meminductor [21].The system has infinite number of equilibrium points and coexisting attractors, and is extremely sensitive to initial values.In the same year, Min et al. added an optimization factor f to a memristive chaotic system, and then obtained a new system without equilibrium point, and realized the control of the hidden attractor [22].In 2020, Wang et al. introduced a memristor feedback into a Lorenzlike chaotic system to obtain a hyper-chaotic system with multistability [23].This system has rich and unique dynamic characteristics.In addition, the nonlinear and random-like behavior of chaotic systems makes them suitable for designing pseudorandom number generator (PRNG).To explore this application, Hua et al. designed a PRNG using a 2D sine chaotification system.It can generate random numbers with high randomness [24].
Up to now, most research works on memristive chaotic systems focus on binary-valued and continuous memristors.Tri-valued or multi-valued memristors can store more information and have richer characteristics than binary-valued memristors.However, tri-valued memristor and its chaotic system haven't received much attention.Therefore, it is of great significance to enrich the characteristics of chaotic systems using the tri-valued memristors.In this paper, we first introduce a tri-valued memristor model, and then propose a four-dimensional hyper-chaotic systems (4D-HCS) using the tri-valued memristor.Theoretical analysis and experimental results show the complex dynamics behaviors of the developed 4D-HCS.
This paper is organized as follows.Section 2 introduces the tri-valued memristor model and its equivalent circuit.Section 3 presents the constructed 4D-HCS, calculates its Lyapunov exponent and Lyapunov dimension, and discusses its stability and equilibrium points.Section 4 analyzes the hidden and coexistent attractors of the 4D-HCS.Section 5 compares the 4D-HCS with the binary-valued memristorbased chaotic system.Section 6 designs a hardware circuit platform for the 4D-HCS and applies it to the design of PRNG.Finally, section 7 gives the conclusions about this paper.

The tri-valued memristor model
The memristor is used to describe the relationship between the charge q and the flux ϕ.According to the definition in [25], ideal memristors can be divided into current-controlled memristors and voltage-controlled memristors, which can be described by Eqs. ( 1) and (2), respectively: where v and i represent the voltage across and current flowing through the memristor, q and ϕ mean the charge and flux on the memristor at time t, R(q) and G(ϕ) are the memristance and memductance of the memristor.
In [26], Wang et al. proposed a voltage-controlled trivalued memristor mathematical model.It satisfies the memristor theory and is different from binary-valued and continuous memristors.The q-ϕ relationship of the model is described by an asymmetric piecewise linear function, whose general expression is shown in Eq. ( 3): where a 0 , b 0 , c 0 , d 0 and e 0 are non-zero constant parameters and c 0 is positive.Taking the derivative of the flux ϕ form Eq. ( 3), the G-ϕ relationship of the tri-valued memristor is calculated as: where sgn(x) represents the symbolic function.When x>0, sgn(x)=1 and when x<0, sgn(x)=-1.
In this paper, we set a 0 =2.5, b 0 =4, c 0 =1, d 0 =2.5 and e 0 =-1.5, according to Eqs. ( 3) and (4), the ϕ-q curve and ϕ-G curve of the tri-valued memristor with three stable memductance 1S, 9S and 4S can be obtained and they are shown in Fig. 1.We can see that the ϕ-q curve is across the origin, and the three different slopes in the curve indicate three memductances of the memristor controlled by the voltage.By applying a sinusoidal signal v(t)=v 0 sin(2π f t) to the above model, and taking amplitude v 0 =4V, frequency f = 0.159Hz, initial value ϕ(0)=-1.5, the v-i curve and timing diagram of the memristor can be obtained and shown in Fig. 2. The circuit characteristics of the memory element appear as a pinched hysteresis loop.Since the multiple pinch-off point behavior reflects the complex nonlinearity of a memristor, it is an important indicator to analyze the characteristics of memristors [27,28].
When applying a sinusoidal signal v(t)=v 0 sin(ωt) to the tri-valued memristor, we can obtain the flux ϕ by Eq. ( 2) as: It can be seen from Eq. ( 5) that the value of the flux ϕ is determined by the initial value ϕ(0), input voltage amplitude v 0 and frequency ω.The maximum flux ϕ S under the sinusoidal voltage input is obtained in half the input cycle, as shown in Eq. (6).
For the ϕ-q relationship of the piecewise linear memristor, its piecewise points ϕ 1 =-1 and ϕ 2 =1 show its nonlinearity, which is affected by the flux ϕ S and the result is expressed as the number of pinch-off points.
It can be seen that the frequency ω affects the value of the flux ϕ S and further changes the nonlinear characteristics of the memristor.The experimental results show that when all piecewise points ϕ 1 and ϕ 2 are included between ϕ(0) and ϕ S , the ϕ-q curve has three pinch-off points and the memristor has high nonlinearity, which is a tri-valued memristor.

Equivalent circuit of the memristor
According to Eqs. ( 2)-( 4), the equivalent circuit of the trivalued memristor can be designed and it is shown as Fig. 4, where the A 1 is an analogue multiplier, U 4 and U 5 are voltage comparators, and the other components are operational amplifiers.
As can be seen from Fig. 4, R 1 , C 1 , and U 1 form an inverting integration circuit, which turns the input voltage v into the negative flux -ϕ.Through the inverting addition circuit composed by R 2 , R 3 , R 4 , and U 2 , ϕ+V 1 can be obtained as the output of U 2 .Similarly, the output at U 3 is ϕ-V 2 and the output of U 4 and U 5 are sgn(•) in Eq. (4).Take U 4 as an example, when the input of U 4 is ϕ+V 1 and 0, the output will be U sat if ϕ+V 1 >0 and it will be -U sat if ϕ+V 1 <0, where U sat is the saturated output voltage.For simplicity, the output of U 4 can be written as U sat sgn(ϕ+V 1 ).Similarly, the output U 5 is -U sat sgn(ϕ-V 2 ).Since R 8 , R 9 , and U 6 form an invert- ing proportional operational circuit, the output of U 6 is -(R 9 / R 8 )U sat sgn(ϕ+V 1 ).The output of U 7 is (R 11 /R 10 )U sat sgn(ϕ-V 2 ).Since R 12 , R 13 , R 14 , R 15 and U 8 form an inverting addition circuit to add the outputs of U 6 and U 7 and -V 3 , the memductance G(ϕ) finally is obtained from the equivalent circuit can be described as Eq.(7).
where U sat is the saturated output voltage of the voltage comparators and it is set as 13.5V.Set V 1 and V 2 in the circuit as constant c 0 in Eq. ( 4), namely 1V and set V 3 as constant a 0 , namely 2.5V.The circuit parameters determined by Eq. ( 7) are After G(ϕ) is obtained, the current going through the memristor can be achieved by multiply the input voltage with the G(ϕ).As shown in Fig. 5, when the input voltage v(t)=4sin(2ω f t) is given ( f =0.159Hz,ϕ(0)=-1.5),the timing diagram of v, i and G are consistent with the simulation result shown in Fig. 2, which verifies the effectiveness of the equivalent circuit.As a nonlinear device, the memristors can be used to construct chaotic systems.In this paper, a new 4D hyperchaotic system (4D-HCS) is proposed by introducing the voltage-controlled tri-valued memristor model to the Lü system.The expression of the 4D-HCS is shown as follows: where x, y, z and w are system state variables, and a, b, c and d are constant parameters.G(w) is the memductance of the tri-valued memristor in Eq. ( 4).When setting these parameters as a=40, b=5, c=24.4 and d=50, and these initial values as [x 0 , y 0 , z 0 , w 0 ]=[0.01,0.01, 0.01, 0.01], the Lyapunov exponents of the 4D-HCS are calculated as LE 1 =4.2486,LE 2 =0.0025,LE 3 =-0.004and LE 4 =-24.8471,and its Lyapunov dimension is calculated as D L =3.1709 using Eq. ( 9).As a contrast, the Laypunov exponents of the Lü system are LE 1 =2.7931,LE 2 =0.0049,LE 3 =-23.3979and its Lyapunov dimension is D L =2.1196.It is obvious that the 4D-HCS has a higher maximum Lyapunov exponent and Lyapunov dimension than the Lü system.
The phase diagrams of the 4D-HCS under the above conditions are shown in Fig. 6, which indicates the rich dynamic behaviors of the system.

Dissipative property analysis
To study the dissipative property, we calculated the divergence of the system using Eq.(10).

Equilibrium point and stability analysis
To analyze the stability of the 4D-HCS, the equilibrium point of the system must be calculated.Let the right part of Eq. ( 8) equal to zero, and the system parameters a, b, c and d are non-zero values, we can obtain an equilibrium point O={x=y=z=0, G(w)=0}.However, the tri-valued memristor model defined by Eqs. ( 3) and ( 4) shows that the memductance G(w) =0, which indicates that the 4D-HCS based on the tri-valued memristor has no equilibrium point.Thus, the chaotic attractors generated by the 4D-HCS are all hidden attractors.

Timing diagram and Poincaré mapping analyses
The Timing diagram of the 4D-HCS is shown in Fig. (8), which indicates that the 4D-HCS shows pseudorandom and aperiodic behaviors.Fig. 9(a) and Fig. 9(b) show the Poincaré mapping on the x-z plane with y=-10 and the x-y plane with z=10, respectively.As can be seen, the Poincaré mapping are composed of dense points with hierarchical structures.
4 Dynamical Property Analysis

Influence of system parameters on dynamic characteristics
The behaviors of a chaotic system are extremely sensitive to the change of its parameters.Therefore, it is necessary to study the influence of parameter variation on system dynamics.

Influence of the parameter a
For the 4D-HCS, we first study its behaviors when its parameter a increases within [1,12] by fixing the initial values [x 0 , y 0 , z 0 , w 0 ]=[0.1,0.1, 0.1, 0.1], and its other parameters b=-1, c=-1.2 and d=50.The corresponding Lyapunov exponent spectrum and bifurcation diagram can be obtained and shown in Fig. 10 and Fig. 11, respectively.It can be seen from Fig. 10 that with the increment of parameter a, the 4D-HCS has different Lyapunov exponents, and thus it exhbits different states.The bifurcation diagram in Fig. 11 shows that the 4D-HCS transforms from the periodic state first to the chaotic state, and then to the hyperchaotic state, as the parameter a increases.The specific evolution process is shown in Table 1, and the representative phase diagrams in x-y plane are shown in Fig. 12.     which is consistent with its Lyapunov exponent spectrum in Fig. 13.Besides, Fig. 15 shows the phase diagram of the attractor on the x-z plane when the parameter c changes.To further study the influence of parameters to the system state, we illustrate the dynamical map of the 4D-HCS.The dynamical map reflects the different dynamic characteristics of a system with the change of multiple parameters.The two-dimensional dynamic map of the 4D-HCS for parameters a and c is shown in Fig. 16, in which the green area It can be seen that when the parameter c is less than -1 and a changes within [10,15], the 4D-HCS mainly shows hyperchaotic behaviors; when c is greater than -1 and a is set as a larger value, the system mainly shows chaotic behaviors; when c is close to zero and a is set as a smaller value, the system mainly show periodic behaviors.

Influence of memristor parameter on dynamic characteristics
The tri-valued memristor improves the dimension of the chaotic system and enlarges the number of parameters of the system.
In this subsection, we analyze the influence of the memristor parameter to the dynamics of the 4D-HCS.
The Lyapunov exponent spectrum and bifurcation diagram of variable x can be calculated with the change of parameter a 0 and they are shown in Fig. 17 and Fig. 18, respectively.
Fig. 17 shows that when a 0 increases within [2.5, 5], the system shows hyper-chaotic behaviors, and Fig. 18 shows that the system changes from the hyper-chaotic (and chaotic) state to the periodic state through inverse bifurcation with the increase of the parameter a 0 .Obviously, when the parameter a 0 varies within [2.5, 30], the memductance of the memristor changes makes the system state changes.This shows that the parameter space of the 4D-HCS is expanded, and its system state is affected by the memductance of memristor.The 4D-HCS still has no equilibrium point when G(ϕ) is positive, and also shows the hidden oscillation phenomenon.
Keep other conditions unchanged and calculate the Lyapunov exponent spectrum and bifurcation diagram of variable x with the change of parameter d 0 and show them in Fig. 19 and Fig. 20, respectively.It can be seen that with the change of parameter d 0 , the 4D-HCS changes from the hyper-chaotic (and chaotic) state to the periodic state through the inverse bifurcation, which is similar to the evolution of the system with the change of the parameter a 0 .

Dynamical map with varying a 0 and d 0
The dynamical map of the 4D-HCS for parameters a 0 and d 0 is shown in Fig. 21, in which the green area marked with P represents the periodic state, the blue area marked with C represents the chaotic state, and the dark blue area marked with H represents the hyper-chaotic state.Fig. 21 shows the different dynamic characteristics of the system with the change of the parameters a 0 and d 0 .Specifically, when the parameters a 0 and d 0 are both small values, the 4D-HCS  In this section, we analyze the power spectrum of the 4D-HCS to further demonstrate the evolution of its state.The power spectrum is obtained by performing the Fourier transform to the output sequence of the 4D-HCS, and it can be used to distinguish the difference of dynamic states.It is well-known that the power spectrum of a periodic signal is a discrete spectrum, and the power spectrum of an aperiodic signal is a continuous spectrum.For an aperiodic signal, its power spectrum is continuous and there are a large number of peaks in its corresponding power spectrum, this is mainly caused by a large number of period-doubling bifurcations in chaos.In addition, for a noise signal, its power spectrum is continuous and smooth.
Fig. 22 shows the phase diagram of different attractors in the y-z plane and the corresponding power spectrum when set the system parameters as a=12, b=-1, d=50.Fig. 22(a) shows the period 1 attractor with c=7, and the corresponding power spectrum has a peak in Fig. 22(b).Fig. 22(c) shows the period 2 attractor with c=6.35, and the corresponding power spectrum has two peaks in Fig. 22(d).Obviously, the power spectrum of the above two periodic signals are all discrete spectrum.Fig. 22(e) shows the hyper-chaotic attractor with c=-2, and the corresponding power spectrum is shown in Fig. 22(f), which is a continuous spectrum, and a large number of peaks are formed due to period-doubling bifurcation.
Therefore, power spectrum analysis can effectively distinguish and compare the periodic signal, chaotic signal and noise signal of the system.

Initial sensitivity
A chaotic systems is also sensitive to the changes in its initial values.With slightly different initial values, the chaotic system can produce completely different sequences.The sensitivity of a chaotic system can be analyzed by measuring the correlation the two sequences generated by different initial values.The correlation coefficient which is defined by Eq. ( 13): where, X t and Y t are two sequences generated by the system under two slightly different initial values, µ and σ represent the mean value and standard deviation of the sequence, E[•] is the expectation function [31].A C o value closer 0 indicates the lower correlation between the two sequences, and further means the higher sensitivity of the chaotic system to its initial values.
For the 4D-HCS, we fix the system parameters [a, b, c, d]=[12, -1, -2, 50], and slightly change each variable in [x 0 , y 0 , z 0 , w 0 ] with a 10 −8 difference.Taking the variable x 0 as an example and let x 0 ′ =x 0 +10 −8 , then the system generates a different sequence pair (X 1 , X 2 ) under initial values [x 0 , 0.1, 0.1, 0.1] and [x 0 ′ , 0.1, 0.1, 0.1].In the same way, slightly change the y 0 , z 0 and w 0 to get (Y 1 , Y 2 ), (Z 1 , Z 2 ) and (W 1 , W 2 ).Then calculate the correlation values of these sequence pairs and their results are shown in Table 3.It can be seen that the correlation values of all the sequence pairs are very close to 0, which indicates that the 4D-HCS is extremely sensitive to changes of its initial values.To visually show the sensitivity, we plot these sequence pairs (X 1 , X 2 ), (Y 1 , Y 2 ), (Z 1 , Z 2 ) and (W 1 , W 2 ) in Fig. 23.It clearly shows the difference between each sequence pair generated by the system with slightly different initial values.Therefore, the 4D-HCS is extremely sensitive to its initial values.

Coexistent attractors analysis
The 4D-HCS not only has hidden attractors, but also coexistent attractors.Under symmetrical initial conditions, the phase diagrams of the coexistent period attractors and the coexistent chaotic attractors of the system are shown in Fig. 24 and Fig. 25, respectively.
In Fig. 24 and Fig. 25, the red area represents the trajectory with initial values [-0.1, -0.1, 0.1, 0.1], while the blue Fig. 23 The timing diagrams of the sequences under the case of As a result, under the symmetrical initial values [±0.1, ±0.1, 0.1, 0.1], a total of five types of coexistent attractors appear when the parameter c changes, as shown in Table 4.

Comparison with the Binary-valued Memristor-based Chaotic System
In this section, we compared the 4D-HCS with binary-valued memristor-based chaotic system to further illustrate the advantages of the tri-valued memristor.

The binary-valued memristor model
In [14], Itoh etal.proposed a binary-valued memristor model, whose q-ϕ relationship is described by a symmetric piecewise linear function, and its general expression is shown in Eq. ( 14).
where α, β and δ are positive parameters.The corresponding memductance G ′ can be described by Eq. (15).

Simulation and comparison
Next, we introduce the G ′ (ω) to replace the G(ω) in Eq. ( 8) to obtain a new system as shown in Eq. (16).
Table 5 compares the simulation results of Eq. ( 8) and Eq. ( 16).It can be seen that under the same system parameters and initial conditions, both systems have hidden attractors.In addition, the system based on the tri-valued memristor has the unique characteristics of hyper-chaos and coexistence attractors, which indicates that the tri-valued memristor has a great advantage in enhance the chaos complexity.

Hardware Implementation and Application
In this section, we implement the hardware circuit platform of the 4D-HCS and apply it to design PRNG.To further verify the easy implementation of the 4D-HCS, we use an improved modular circuit design scheme to construct the hardware circuit of the system.In numerical simulation, the range of the system variables is much larger than the linear dynamic range (±13.5V) of the operational amplifier.Therefore, to avoid nonlinear distortion, the proportional compression transformation and time scale transformation must be performed first.For Eq. ( 8), let kx→x, ky→y, kz→z, kw→w and t=τ 0 t, where k=50 and τ 0 =100.Determine the parameters a=12, b=-1, c=-1.2 and d=50, the final system can be described by the following equations: The design result of the modular hyper-chaotic circuit is shown in Fig. 28, and the state equations of the circuit can be described by: According to Eqs. ( 17) and ( 18), the parameters of the circuit can be determined as: In addition, the operational amplifiers TL084CN(U 1 -U 6 ) and analogue multipliers AD633(A 1 -A 2 ) with ±15V power are used to construct the circuit.Fig. 29 shows the physical hyper-chaotic circuit, and Fig. 30 shows the experimental results.Comparing the experimental results with the simulation results in Fig. 7, we can observe that the theoretical analysis is consistent with the physical experiment.This further proves the reliability of the 4D-HCS.

Application in pseudorandom number generator
To further investigate the complex dynamics behaviors of the 4D-HCS, we further designed a PRNG to analyze the performance of its hyper-chaotic sequences.First, set the parameters a=12, b=-1, c=-1.2 and d=50.For the initial values x 0 , y 0 , z 0 within [0.1, 1] along w 0 =0.1, we take an interval 0.1 to obtain 1000 combinations of initial values.Then generate hyper-chaotic sequences using the Runge-Kutta methods.The time length is set as 135s, and the step size is set as 0.001.Remove the first 10000 numbers, and finally get 1000 sets of different hyper-chaotic sequences X i , Y i , Z i , W i (i=1, 2, . . ., 1000), each of which has a length of 125000.
Next, a PRNG is designed to generate pseudorandom numbers (PRNs) and it is defined as: A PRNG is expected to generate PRNs with high randomness.Here, we use the National Institute of Standards and Technology (NIST) SP800-22 test suit in [32] to test the performance of the pseudorandom bitstreams generated by the proposed PRGN.The NIST SP800-22 test includes 15 subtests.For each subtest, a P-value is generated and the sequence is considered to pass the subtest if the P-value is greater than the significance level α, which is usually set as 0.01.
Here, we choose the hyper-chaotic sequence sets X and Z as two examples to generate two binary stream sets with a sample size of 1000 and a sample length of 10 6 , which meet the test conditions described in [32].The test results are given in Table 6, where P-value T represents the uniform distribution of P-values, which is calculated as: where igamc(•) is the incomplete gamma function and: where the range of P-value is evenly divided into 10 subintervals, F i represents the number of P-value in the ith subinterval, and s represents the sample size.If the obtained Pvalue T ≥0.0001, the sequences can pass the corresponding subtest.And the minimum pass proportions are approximately 0.980 for a sample size 1000.The test results in Table 6 show that all P-value T are larger than 0.0001 and all pass proportion are larger than or equal to the minimum pass rate of 0.980, indicating that the obtained PRNs can pass all the 15 subtests.This indicates that the proposed PRNG using the 4D-HCS can generate a large number of PRNs with high randomness.Therefore, the 4D-HCS system has potential application in the field of cryptography.

Conclusion
This paper proposes a new 4D-HCS based on the tri-valued memristor.First, the mathematical model of memristor is analyzed and its equivalent circuit is verified.Then, the 4D-HCS is constructed by introducing the memristor into an existing chaotic system.Stability analysis results show that the  4D-HCS has no equilibrium point and thus has hidden attractors.The dynamic characteristics of the 4D-HCS are analyzed in aspects of the Poincaré mapping, bifurcation diagram, Lyapunov exponential spectrum and power spectrum.The analysis results verify that the introduction of tri-valued memristor can greatly enhance the chaos complexity of existing chaotic system.Besides, the characteristics of the coexistent attractors of the 4D-HCS are analyzed, and the results proves that the 4D-HCS can show five types of coexistent attractors.Furthermore, we compared the simulation results of the binary-valued memristor-based chaotic system.The results show that the introduction of the tri-valued memristor is beneficial to the system to produce more complex characteristics.To show the easy implementation of the 4D-HCS, we build a physical circuit to implement the system and the implementation results show the correctness of the theoretical analysis.Finally, a PRNG is designed using the 4D-HCS.Test results shows that the PRNG can generate random numbers with high randomness.Therefore, the application of such kind of memristive hyper-chaotic systems in cryptography deserve to further explore in the future.

Fig. 2
Fig. 2 The v-i characteristic curve of the voltage-controlled tri-valued memristor model: (a) Pinched hysteresis loop, (b) Timing diagram

Fig. 4
Fig. 4 Equivalent circuit of the memristor

Fig. 5
Fig. 5 Simulation results: (a) Timing diagram of v and i; (b) Timing diagram of G

Fig. 13
Fig. 13 Lyapunov exponent spectrum corresponding to c

Fig. 14
Fig. 14 Bifurcation diagram corresponding to c

Lyapunov exponent spectrum corresponding to a 0 Fig. 18
Fig. 17Lyapunov exponent spectrum corresponding to a 0

Fig. 21
Fig. 21 Dynamical map of parameters a 0 and d 0

Table 1
Different system states corresponding to different values of a

Table 2
Different system states corresponding to different values of c

Table 3
Correlation values of different sequences Initial values Correlation coefficient Correlation of X 1 , X 2 Correlation of Y 1 , Y 2 Correlation of Z 1 , Z 2 Correlation of W 1 , W 2

Table 4
Each type of coexistent attractors

Table 5
The results of comparison

Table 6
NIST test results of linear congruence and hyperchaotic pseudorandom sequences Non-overlapping template test, Random excursions test, and Random excursions variant test are comprised of 148, 8, and 18 subtests, respectively.The result with the lowest pass proportion of multiple subtests is reported. *