A New Memristor-Based 5 D Chaotic System and Circuit Implementation

1Tianjin Key Laboratory for Civil Aircraft Airworthiness andMaintenance, Civil Aviation University of China, Tianjin 300300, China 2College of Information Engineering and Automation, Civil Aviation University of China, Tianjin 300300, China 3Engineering Technology Department, Sichuan Airlines, Chengdu 610202, China 4Department of Engineering Design and Mathematics, University of the West of England, Frenchay Campus, Coldharbour Lane, Bristol, BS 161QY, UK


Introduction
Memristors are the fourth kind of circuit elements except for resistors, capacitors, and conductors and are conceived by Chua in 1971 through the basic symmetric principle [1].Furthermore, the corresponding theory was applied to memristive devices in 1976 [2].It took a long time to develop the hardware memristor model.Until 2008 HP labs first realized the memristor of nanoscale in the form of crossbar array [3].Since memristors have the potential applications in the wide range of fields, memristor study becomes hotter, and a huge amount of researchers have paid immense attention on memristor studies from industry and academics, respectively [4,5].The typical examples include nonvolatile memories of nanoscale [6], memristor-based synapse in neuromorphic systems [7,8], logic operations through material implication [9][10][11][12], and nonlinear dynamics in chaotic system [13][14][15][16][17].
With rapid development of memristor models, some studies combine the memristor and chaotic systems including dynamics analysis, image encryption applications and circuit implementations which have grown up quickly in recent years [18][19][20][21][22].One of the typical early memristor-based chaotic systems was developed by Itoh and Chua in 2008 [19].This paper developed some nonlinear oscillators by using memristors based on Chua's oscillators.Petráš derived and investigated a fractional-order memristor-based Chua's circuit in [20].Chua and Muthuswamy also discussed circuit topology and developed the simplest memristor-based circuits [21].These papers demonstrate that the memristor oscillators own the special nonlinear dynamics due to memristors' extinguished characteristics.One of these significant characteristics is that the behaviors are dependent on initial states and circuit parameters.Li et al. proposed a scroll chaotic system circuit implementation by using HP memristor [22].Ma et al. developed a four-wing hyperchaotic system by using a memristor adding over a three-dimensional chaotic system [16].Dimitrios et al. found a new 4-D memristive chaotic system and investigated the behavior with hidden attractors of the system through numerical simulations [23].Wang et al. proposed a flux-controlled memristor model and established a 4-D chaotic system with this model.The numerical analysis and circuit implementation simulation verification were conducted [24].Mou et al. discussed the characteristics of dynamical behaviors of a fractional-order 2 Complexity 4D hyperchaotic memristive system and circuit simulation verification [25].Other memristor-based hyperchaotic systems were also investigated such as numerical analysis about a four-dimensional hyperchaotic system with memristor and conducted circuit simulation verifications [26][27][28][29].There are also some other chaotic systems developed by memristive models and its applications [30][31][32].
The above memristor-based chaotic system literatures focus on four or lower-dimensional chaotic systems and investigate the detailed numerical analysis and the corresponding numerical and circuit implementation simulation verifications.However, the higher-dimensional (5-D or above) memristor-based hyperchaotic systems and the corresponding physical hardware experiments are not found.Therefore, the paper analyzes a new memristorbased hyperchaotic system and develops a circuit physical implementation method by using the modularized design method.This method is used to design the circuit without dimensions for chaotic circuit designs and is easy to be implemented in the circuit by using less circuit parts [33][34][35][36][37][38].
The novelty of this paper is to develop a new memristorbased 5D hyperchaotic system, design and analyze the adaptive synchronization of this new system, implement the physical experiment circuit hardware, and verify the existence of system attractors.The improved modularized design method is used to implement the circuit of the system to verify the existence of attractors.
The rest of the paper is organized as follows.Section 2 analyzes the fundamental characteristics of the new memristor-based chaotic system.Section 3 investigates the adaptive synchronization of the new memristor-based hyperchaotic system.Section 4 discusses the circuit implementation of the new system and verifies the existence of attractors.Conclusions are presented in Section 5.

Analysis of a New Memristor-Based Hyperchaotic System
In this section, numerical analyses are conducted for a fluxcontrolled memristor-based new 5D hyperchaotic system derived from Wang's 4D hyperchaotic system.As illustrated in [16], memristor model is based on the fundamental characteristics of a flux-controlled memristor described below.
where  and V are the current and the voltage of the device terminal, respectively.() is the incremental memductance defined as This demonstrates that the characteristics of a memristor are a nonlinear function reflecting the relationship among the charge and flux across and through the device.
Then the memductance is shown below This paper develops a 5D memristor-based chaotic system which is derived from four-wing autonomous chaotic dynamics systems reported by Wang et al. [34,37] This system has four state variables , , , and , and ,  ∈ R + .Substitute (4) into (5), a 5D memristor-based system is obtained.where and , , ,  are positive parameters.
Typically, when  = 14 and  = 78,  1 * is calculated as shown below Two of the five eigenvalues of  1 * are complex conjugates; therefore, it is difficult to determine the stability of the line equilibria.6) is symmetric with respect to  axis since it is invariant when applying the coordinate transformations.

Lyapunov Spectrum and Bifurcation Diagram.
Fix parameters  = 14,  = 78 and vary the parameter , and the graphs about the Lyapunov exponents versus , bifurcation diagram, and phase portraits are shown in Figure 1.
Figure 1 shows that the Lyapunov exponents vary with the parameter  changes.In the five Lyapunov exponents, three of them are obviously negative when  ∈ [0, 5.9].The top two lines in Figure 1(b) demonstrate that these two kinds of Lyapunov exponents are bigger than zero, and systems are hyperchaotic systems when  lies in this range.In this paper,  will be selected in this range.

Adaptive Synchronization of a New Memristor-Based Hyperchaotic System
In this section, the adaptive controller was designed for the new memristor-based hyperchaotic system which was derived by Lyapunov stability theory inspired by [39].Second, consider the slave system shown as below where  푖 ( = 1 ⋅ ⋅ ⋅ 5) are adaptive controller, and  耠 ,  耠 ,  耠 ,  耠 ,  耠 are the new state variables.Therefore, the synchronization errors among the Systems ( 17) and ( 18) are defined as follows.
Then the dynamics of the error can be calculated by substituting the master and slave systems as shown below where the adaptive controller  푖 are defined by where  푖 ( = 1, . . ., 5) are the positive gains and â(), b() are the estimations of the corresponding parameters  and , respectively.Correspondingly, the dynamics of the error is changed into Substitute ( 23) into the dynamics (22), the simplified error dynamics is described as Correspondingly, Depending on the master and slave systems definitions and error dynamics transformations mentioned above, consider a Lyapunov function defined by Differentiate  along the trajectories for equations ( 24)-( 25), V can be obtained as Then, the parameter update law is Theorem 1.The master and slave Systems ( 17)- (18) for a new memristor-based hyperchaotic system along with unknown parameters and positive gains  푖 ( = 1 ⋅ ⋅ ⋅ 5) synchronize exponentially by using the adaptive controller (21) and the parameter update law (28).
Proof.Substitute the parameter update law (28) into the adaptive controller (21), then where It is obvious that () → 0 exponentially as time goes to infinite for all initial conditions of (0).According to Lyapunov stability theory, two systems are synchronized.17): (a) x-y plane with scales 5 V/div and 2 V/div, (b) x-z plane with scales 2 V/div and 1 V/div, (c) x-w plane with scales 2 V/div and 1 V/div, (d) z-w plane with scales 2 V/div and 1 V/div, (e) y-w plane with scales 2 V/div and 1 V/div, and (f) y-u plane with scales 2 V/div and 2 V/div.

A New Memristor-Based Hyperchaotic System Circuit Implementation
In this section, modularized design methods will be implemented in the new memristor-based hyperchaotic systems mentioned above.Multisim software is used to generate the circuit simulation results, and the corresponding physical circuit experiments are conducted to verify the hyperchaotic attractor existences of this hyperchaotic system.
Parameter  represents the strength of a memristor.

𝑘 = 𝑅
In Figure 4, select the following resistors and the capacitor,  7) and part of equation ( 6).
(A2) Memristor-Based Chaotic System Circuit Analysis.The implementation of memristor-based chaotic system uses the Complexity modularized design method applied on System (17).After the time-scale transformation method is applied on System (17), System (17) becomes System (32).

𝑑𝑥
Comparing equations ( 35) to (32), parameters R and C in the circuit can be calculated when using the factor of 0 In the modularized circuit design for System (17), there are five channels for each variable.The circuit implementation uses analog amplifiers LF347N and AD633JN multipliers to implement the addition and the integral operations.Some additional R and C components are also selected in the circuit implementation.
The circuit channels for first four variables are shown in Figure 5, and the fifth one is shown in Figure 3.

(A3) Circuit Implementation for the New 5D Chaotic Systems.
According to the circuit design, the Multisim simulation implementation for the memristor-based hyperchaotic System ( 17) is demonstrated in Figure 6.That shows the attractor phases graphs of different planes for System (17).
Attractor phases of System (17) as shown in Figure 6 own similar hyperchaotic characteristics with those of System (5) which are elaborated in [37].It is known that practical analog devices are not ideal.Therefore, in reality, the voltages added on the amplifiers are not exceeding ±13.5V in order to guarantee the system variable outputs do not exceed the amplifier linear region.This paper employs ±12.5V across the amplifier in the physical circuits hardware, and the corresponding attractor phases graphs are shown in Figure 7.
Comparing Figure 6 to Figure 7, it is shown that the physical experiment attractor phase results match those conducted by Multisim and demonstrate the existence of the memristorbased hyperchaotic attractor for System (17).Furthermore, the memristor-based Wang hyperchaotic system has similar characteristics with those of the original Wang hyperchaotic system which is verified by the Multisim simulation and physical experimental results.The improved modularized method with compression coefficients is flexible, reliable, and straightforward to realize the physical implementation.The memristor-based hyperchaotic system has potential and bright application physically.For example, it can realize the physical communication encryptions more reliably and easily.Future works will focus on the physical applications for communication encryptions by using the memristor-based hyperchaotic system.

Conclusions
The paper developed a new 5D memristor-based chaotic system with a flux-controlled memristor.The dynamics analysis of the system showed that the new system is a hyperchaotic system.The adaptive controller and update laws for the synchronization of the new system were designed and conducted.Furthermore, the modularized design method for the physical circuit experiment implementation is applied to realize the circuit by Multisim and physical experiments hardware.After the implementation of these circuits, then the comparisons between Multisim simulation and results from the physical experiments showed that these two kinds of results matched with each other and verified the existence of the attractors from the phase plane graphs.Furthermore, this new higher-dimensional chaotic system with memristor owns more complex dynamics and can be applied in a wide range of applications such as encryptions.This is also the future work for this work.

Figure 4 :
Figure 4: The relationship between  and () for the circuit as shown in Figure 3.

Figure 5 :
Figure 5: Modularized circuit channels for circuit implementation of the new memristor-based 5D hyperchaotic system.

Figure 6 :
Figure 6: Multisim implementation of the chaotic System (17): (a) x-y plane with scales 5 V/div and 2 V/div, (b) x-z plane with scales 2 V/div and 1 V/div, (c) x-w plane with scales 2 V/div and 1 V/div, (d) z-w plane with scales 2 V/div and 1 V/div, (e) y-w plane with scales 2 V/div and 1 V/div, and (f) y-u plane with scales 2 V/div and 2 V/div.

Figure 7 :
Figure 7: Physical circuit implementation hardware and digital oscilloscope diagrams of the chaotic System (17): (a) x-y plane with scales 2 V/div and 1 V/div, (b) x-z plane with scales 2 V/div and 1 V/div, (c) x-w plane with scales 2 V/div and 1 V/div, (d) z-w plane with scales 1 V/div and 1 V/div, (e) y-w plane with scales 1 V/div and 1 V/div, and (f) y-u plane with scales 1 V/div and 2 V/div.

Figure 4
Figure 4 is the Multisim simulation for the flux-controlled memristor with  input with sinusoid signal.This demonstrates the typical closed loop characteristics of the memristor developed by equation (7) and part of equation (6).
paper employs the unified compression coefficient,  = 1/20, for each variable in order not to exceed the range of the oscilloscope used to the circuit.Then