A Chaotic Oscillator Based on HP Memristor Model

This paper proposes a simple autonomous memristor-based oscillator for generating periodic signals. Applying an external sinusoidal excitation to the autonomous system, a nonautonomous oscillator is obtained, which contains HP memristor model and four linear circuit elements. This memristor-based oscillator can generate periodic, chaotic, and hyperchaotic signals under the periodic excitation and an appropriate set of circuit parameters. It also shows that the system exhibits alternately a hidden attractor with no equilibrium and a self-excited attractor with a line equilibrium as time goes on. Furthermore, some specialties including burst chaos, irregular periodic bifurcations, and nonintermittence chaos of the circuit are found by theoretical analysis and numerical simulations. Finally, a discrete model for the HP memristor is given and the main statistical properties of this memristor-based oscillator are verified via DSP chip experiments and NIST (National Institute of Standards and Technology) tests.


Introduction
A memristor is a two-terminal circuit element, which was defined by Chua in 1971 [1].Until 2008 the memristor with nanoscale was first realized in a TiO 2 crossbar array structure form by Williams group from HP (Hewlett-Packard) labs [2].As a new device with memory, its potential applications span different fields including nonvolatile memories on the nanoscale [3,4], neuromorphic systems [5], chaotic circuits [6], and others [7,8].Thus the HP memristor has attracted immense interest rapidly from both industry and academia [9].
In many studies, applications of memristor are one of the active topics of research, since memristor is a new nonlinear memory element and can be used in memristorbased oscillator, information encryption, memory, and so on.
The first memristor oscillator with piecewise linear (PWL) - characteristics was proposed by Itoh and Chua [10], which was obtained from Chua's circuit.Then some memristor oscillators with a PWL nonlinearity or a smooth piecewise-quadratic nonlinearity are presented based on Chua's circuit [11,12].Meanwhile, some other memristor oscillators based on Chua's circuit are proposed, using diverse memristor models with the - nonlinearities: () =  +  3 [13].In these studies, some new properties, which are different from conventional chaos, such as equilibrium set, transient chaos, and stable chaos with an intermittence period, are found.
On the other hand, a few nonautonomous memristor oscillators are proposed.In [13], an oscillation circuit based on van der Pol circuit oscillator is implemented.Very recently, nonautonomous Chua's circuit is constructed by replacing Chua's diode with a memristor model characterized by the memductance () = − + |()|, which can exhibit some complex dynamical properties including transient chaos, transient hyperchaos, and chaotic beats [14].Now the most common nonlinear - characteristics, including PWL, cubic, piecewise-quadratic, and absolute value function nonlinearity, of memristor are assumed and used to design memristor circuit.But HP TiO 2 memristor is a realized operational device, so it is of great significance for researching application circuits with the HP memristor.However, only a few studies on HP memristor-based oscillators have reported.In [6], a chaotic oscillator based on the HP TiO 2 memristor is introduced, which is very important for practical applications of memristor.But this circuit makes use of two HP memristors in parallel with opposite polarities, thereby showing a defect for circuit complexity.Reference [15] presents a complete model of HP memristor, which takes into consideration the interdependence between memristance, charge, and flux along with the boundary and initial conditions of operation.And a scroll chaotic system containing the HP memristor model and triangular wave sequence is proposed in [16], which is only a mathematical system consisting of three differential equations and contains two nonlinear functions (one of them is memristor model), showing a complexity for its mathematical structure and implemented equivalent circuit.In [11], a new chaotic system, based on the flux-controlled memductance model of the HP TiO 2 memristor and Chua's circuit, is designed.
In order to generate more complex pseudorandom signals, this paper designs a nonautonomous oscillator using the mathematical model of flux-controlled memristor, which can generate periodic signals, chaotic and hyperchaotic signals, thus exhibiting more complex dynamics.In order to verify the physical realizability of the proposed oscillator, a scheme for digitally realizing this memristor oscillator is provided by using the DSP chip, and the statistical property of the chaotic oscillator is tested by using the NIST test suite.NIST test results show that randomness of the proposed oscillator is better than that of Lorenz system and is definitely up to the standards of the NIST.
The DSP technology is used to realize this memristor system, since DSP is one of mainstream techniques for information processing at present.Moreover, DSP can adapt to practical applications of both chaotic PN sequence generation and chaotic information encryption/decryption processing.To this end, a discrete model of HP memristor is given first.

Flux-Controlled Memristance of HP Memristor
In the seminal paper which first announced the fabrication of TiO 2 memristor, the memristance is the charge-controlled form.For convenience, sometimes it is very necessary to give the flux-controlled memristance of HP memristor.Figure 1 shows the structure of HP TiO 2 memristor, which is described by [2] where  off and  on are the high resistance and low resistance for  = 0 and  = , respectively,  V denotes the dopant mobility, () is the charge across the memristor, and () is the memristance.Other variables and parameters are shown in Figure 1.Now, we try to find the relation between () and flux () in the memristor for obtaining the flux-controlled memristance.To this end we can rewrite equivalently (2) as follows: where  on = V on /() is the field intensity in the doped region of the memristor and V on is the voltage across the region.Namely, V on is the voltage across the resistance,  off ()/.Thus, V on can be written as From ( 5) and ( 4), we can get the following equation: By integrating (6) with respect to time , the relation between () and () is described as where () = ∫   0 V().By solving (7), the relationship between () and flux () can be obtained as follows: By inserting (8) into (3) the flux-controlled memristance of this memristor is obtained, which is simplified to where  1 = − on  off /( off −  on ),  2 =  4 off /( off −  on ) 2 , and  3 = 2 on  2 off  V / 2 ( off −  on ).In (9), because  off ≫  on ,  1 is negative and  2 and  3 are all positive; therefore () has both positive and negative values, corresponding to both passive and active memristors, respectively.By using the HP memristor model, we can design simplest autonomous oscillator for generating periodic or chaotic oscillations.
Figure 2 shows a two-terminal circuit, which consists of a passive HP memristor and a negative conductance (or a negative resistance).The equivalent memductance of this two-terminal circuit is given by where When  > 1/(), this two-terminal circuit becomes an active memristive system; that is, () = 1/() −  < 0, implying that the circuit can be regarded as an equivalent active memductance (or an equivalent active memristance) shown in Figure 2(b).

An Autonomous Memristor Oscillator
A simple autonomous memristor system is designed as in Figure 3(a), which consists of a resistor , an inductor , a capacitor , a negative resistor −, and active flux-controlled memductance .
If we, respectively, change the inductor parameters  and  from 2 H and 0.891 mS to 2.65 H and 0.791 mS while maintaining all other circuit parameters unchanged, taking Ω = 850 Hz and  = 14.15V, the corresponding parameters of system ( 14) are  = 0.378,  = 5.35,  = 0.085,  1 = −0.2, 2 = 2.08, and   3 = 0.039.Under the conditions of these parameters, this system exhibits a chaotic attractor, as shown in Figure 5.
Notice that Figure 5(d) shows the chaotic V  −   hysteresis curves of the flux-controlled memristor, which contains a lot of nonrepeating V  −   curves, thereby demonstrating chaotic properties of this system.

Basic Dynamical Characteristics.
From system (14), one has Therefore, to make system (14) dissipative, it must satisfy that is, in order to generate chaotic attractors, the total resistance for parallel connection of the negative resistor and the memristor in Figure 3 must be less than the value of one  of the linear resistor .To this end, the system must meet  < 122.6 for the above fixed system parameters.Strictly speaking, the equilibria of system ( 14) are related with excitation voltage.When sin Ω ̸ = 0, that is, Ω ̸ =  ( = 0, 1, 2, . ..), this system is written as where V ̸ = 0 and V is an instantaneous value of the excitation voltage at time  ( ̸ = /Ω).Let ẋ = 0, ẏ = 0 and ż = 0; (17) has no solution of equilibria, and then it becomes a system with no equilibria, which is possible to display hidden attractors without any equilibria [9].
When sin Ω = 0, that is, Ω =  ( = 0, 1, 2, . ..), or the external excitation equals zero, that is, sin Ω ≡ 0, (14) becomes a three-dimensional autonomous system, which corresponds to (12).At the moment, it has an equilibrium set, which is obtained by solving the equations ẋ = ẏ = ż = 0: where  is a real arbitrary constant, thereby implying that these equilibria occupy the whole -axis, which is also called a line equilibrium [9].Considering the above two cases together, that is, Ω ̸ =  and Ω =  ( = 0, 1, 2, . ..), we can see that the system switches alternately between a nonautonomous subsystem with no equilibrium and an autonomous subsystem with a line equilibrium under the control of sine voltage excitation.Therefore a hidden attractor and a self-excited attractor appear exchangeably with the time evolution.
Particularly, when the external sine excitation is removed from the nonautonomous system, that is, sin Ω ≡ 0, it will become the autonomous system shown in Figure 3.
On the other hand, by solving (20), we obtain the eigenvalues of Jacobian matrix (19): Figure 6 shows the eigenvalues with respect to the equilibrium , which can determine stability of the system.When  < −52,  is an unstable saddle point with a positive real root, a negative real root, and a zero root.With the increasing of , absolutes of the two real roots increase quickly and reach a maximum at  = 52 and then become two complex conjugate roots with the same negative real components when  ≥ −52.At  = 123.64, the two real components become positive, yielding unstable focus.

Chaos and Hyperchaos in Bifurcations.
The dynamics of ( 14) depend on the parameters , , and  and other circuit parameters.Now we begin to explore the dynamical characteristics varying with these parameters.To start with, we observe the bifurcation diagram and Lyapunov exponent spectrum with respect to , as shown in Figure 7, where  = 0.378,  = 0.085,  1 = −0.2, 2 = 2.08, and  3 = 0.039.By increasing the amplitude  from zero, the system exhibits the bifurcation route from periodic orbits without positive exponent to chaos with one positive exponent, followed by hyperchaos with two positive exponents.In particular, a novel phenomenon occurs at  = 4.5, where a period 3 orbit and an irregular period bifurcation appear but without obvious period-doubling bifurcations.Figure 8 shows the corresponding hyperchaotic, chaotic, and periodic orbits.
Figure 9 shows the bifurcation diagram and the corresponding Lyapunov exponent spectrum with respect to , in which burst chaos occurs at  = 0.082 and the system undergoes complex bifurcations from chaos to complex periodic orbits to chaos, where irregular period bifurcation occurs again.
Figure 10 shows the bifurcation diagram and corresponding Lyapunov exponent spectrum for  with respect to the parameter  related with the circuit parameters , , and .From Figure 8 we can see that with the increase of , the system also exhibits a complex evolution law from period to chaos to hyperchaos, and it eventually evolves to a periodic orbit.

Oscillator with a Nonlinear Drift Model of Memristor
The memristor model described in ( 1)-( 4) is a linear drift model.However, according to the experimental observation in the HP memristor, the drift speed / is nonlinear for the current () through the memristor.So we describe the real memristor by using the following nonlinear drift model [6]: where  = /; stp() = 0 for  < 0 and stp() = 1 for  > 0 or  = 0.If we take the nonlinear drift model of (25), the state equations of the circuit shown in Figure 4 are described by When  = 0.205,  = 0.75,  = 1.5,  = 0.2 mS,  on = 0.9 kΩ,  off = 2.5 kΩ,  = 3.0 nm, and  V = 10 −14 cm 2 s −1 v −1 , this oscillator with a nonlinear drift model produces a chaotic oscillation and its attractors are shown in Figure 11, where the Lyapunov exponents of this oscillator are  1 = 0.115,  2 = 0, and  3 = −0.114.

Digital Realization of the Memristor System
Now, we provide a scheme for digitally realizing the proposed flux-controlled memristor system.
The DSP technology is used to realize this memristor system, since DSP is one of mainstream techniques for information processing at present.Moreover, DSP can adapt to practical applications of both chaotic PN sequence generation and chaotic information encryption/decryption processing.To this end, a discrete model of HP memristor is given first.
According to (1)-( 3), the discrete model of memristance can be obtained: where ∑  =−∞ (  )Δ  =   .Hence, from (28) and ( 3)-( 8) the discrete model of the flux-controlled memristor is derived: By applying Euler's method to (14), the discrete equations of the memristor system are described as Based on the DSP evaluation board (ICETEK-VC5509-AE-sampling frequency is 5 M/S and 8 bits' quantization) and the software platform (CCStudio v3.3), the flux-controlled memristor system is experimentally realized.Figure 12 shows the experimental orbits of period and chaos, observed by an analog oscilloscope, and the hyperchaotic orbits are shown in Figure 13.Compared with Figures 5 and 8, these experimental orbits are in good agreement with the Matlab simulation results.
Based on the DSP technology, digital pseudorandom sequences, or digital chaotic sequences, can be obtained by using the proposed memristor oscillator, which can be used in various applications of information safety.Figure 14 shows the digital pseudorandom sequence generated by experimental DSP evaluation board and observed by a digital oscilloscope, in which upper wave is the clock pulses and the lower wave is the chaotic pseudorandom sequences.Now, we evaluate the randomness of the proposed memristor oscillator by testing its binary sequences and using the NIST test suite and compare the randomness with that of the well-known Lorenz system.
The NIST test suite is the most authoritative tool for pseudorandom test currently, which contains 15 tests.The 2.0 version of the test suite package is used in this paper.For a binary sequence, which is generated by the proposed memristor oscillator and contains "0" and "1" of a given length , this sequence is divided into  nonoverlapping parts where each length is  ( = /), where  = 1,000,000,000 and  = 1000.
An analysis report is generated by the test suite package, which consists of some relevant intermediate values, including proportions and  values for each test.The analysis report is shown in Table 1.For comparison, the analysis report of the well-known Lorenz oscillator is shown in Table 2.
From Table 1 we can see that all of the statistical test's  values pass the test.However, for the well-known Lorenz oscillator the tested item "Overlapping Template" fails to pass the test.
Moreover, from Tables 1 and 2, although both the memristor oscillator and the Lorenz oscillator all pass the tests of proportion, all the tested proportion items of memristor oscillator are greater than that for the Lorenz oscillator except for the item "Block Frequency." Therefore, we can conclude that the random characteristic of the memristor oscillator is better than that of the wellknown Lorenz oscillator.

Conclusion
In conclusion, this paper has designed a nonautonomous flux-controlled memristor-based oscillator system by using the mathematical model of practical HP TiO 2 memristor.This proposed system possesses many specialties: (i) the system exhibits complex bifurcation from period to chaos and then to hyperchaos; (ii) when external excitation voltage is zero, this nonautonomous system degrades into autonomous system with a periodic oscillation; (iii) this system has some special phenomena, such as burst chaos, and irregular bifurcation started from period 3 but without period-doubling bifurcations; and (iv) it has no transient chaos or intermittence chaos which is an advantage for generating continuous chaotic sequences.In addition, DSP experiment shows that DSP technology can be used to realize the chaotic memristorbased system and in realistic applications of information processing for generating PN sequences and performing various operations such as encryption and decryption.Actually, only using a mathematical model of the HP memristor rather than the actual memristor device, we can build an oscillator and realize a pseudorandom sequence generator based on DSP chip, which provides great convenience for designing and implementing a memristor-based oscillator in actual applications.
The statistical property of the memristor-based chaotic oscillator has been tested by using the NIST test suite.Tested results show that random characteristic of the oscillator is definitely up to the standards of the NIST and is better than that of the well-known Lorenz system.Therefore, this memristor-based oscillator can be used to design pseudorandom sequence generators for various applications of information safety filed.

Figure 14 :
Figure 14: Experimental chaotic sequences generated by the proposed memristor-based oscillator circuit.

Table 1 :
Depiction of the test report for memristor oscillator.

Table 2 :
Depiction of the test report for Lorenz oscillator.
*is the tag that meaning didn't pass the test item.