Memristive Computation-Oriented Chaos and Dynamics Control

A variable boostable chaotic system and the Hindmarsh–Rose neuron model are applied for observing the dynamics revised by memristive computation. Nonlinearity hidden in a memristor makes a dynamic system prone to be chaos. Inherent dynamics in a dynamic system can be preserved in specific circumstances. Specifically, as an example, offset boosting in the original system is inherited in the derived memristive system, where the average value of the system variable is rescaled linearly by the offset booster. Additional feedback from memristive computation raises chaos, as a case, in the Hindmarsh–Rose neuron model the spiking behavior of membrane potential exhibits chaos with a relatively large parameter region of the memristor.

In this paper, a memristor-based computation is introduced in a variable boostable chaotic system and the Hindmarsh-Rose neuron model for observing the influence of memory from the memristive computation. As shown in Figure 1, chaos oscillation, and dynamics control are explored and discussed in a variable boostable system and neuron cells. In A 3D Memristive Chaotic System, the VB14 system is applied to host a memristor, where chaotic oscillation is preserved with the inherent property of offset boosting. The system and component parameter shows their separate power for preserving chaos and revising dynamics . In A Memristive Hindmarsh-Rose Neuron, the Hindmarsh-Rose neuron model is studied as an example to demonstrate the effect of memory. Memristive computation shows its disturbance for chaos production. Some discussion and future work are presented in the last section.

Basic Analysis
A variable boostable chaotic system is a class of chaotic systems with an independent variable that can be controlled with a linearly rescaled average value. By introducing a constant, the offset of the variable can be boosted to any desired level. Thus, we can control the variable switch between a bipolar signal and a unipolar signal according to the constant. In this paper, VB14 is selected as the seed system for hosting a memristor, (1) when a 3.55, b 0.5, IC (1, 0, 1), system (1) is chaotic with Lyapunov exponents (0.151, 0, −0.651) and Kaplan-Yorke dimension D KY of 2.2319. A memristor is introduced into the VB14 system, as shown in Eq. 2, where the flux-controlled memductance is described as, (3) Here y is the internal variable of the memristor and can be regarded as the flux-controlled variable of the memristor. The memristor function curve and the corresponding hysteresis loop are plotted as shown in Figure 2. In the process memristor with the internal variable determined as, _ y z 2 − yz is selected for introducing a memristor-based computation. Here x, and z are system variables, a, and b, are the bifurcation parameters of the system (2); and c, and d are the internal parameters of the memristor. When a 2.4, b 0.5, c 1.7, and d 2.5, under initial conditions (1, 0, 1), system (2) has a chaotic attractor of Lyapunov exponents (0.18548, 0, −1.2907) and Kaplan-Yorke dimension of 2.1437, as shown in Figure 3.
The hypervolume contraction of this system can be described by the following expression: which is related to the values of the system variable z and parameter b. When system parameters are a 2.4, b 0.5, c 1.7, and d 2.5, the result of this equation ∇V is negative. Further examination revealed that the sum of Lyapunov exponents is also negative. So, system (1) is dissipative.

Bifurcation Observation
For system (2), the dynamic behavior of the system can be modified by changing the parameters. When b 0.5, c 1.7, d 2.4 and the initial condition is (1, 0, 1), the bifurcation diagram and Lyapunov exponents of parameter a varies in [0.5, 3] are shown in Figures 4A,B. It can be seen from the figure that the system has a couple of periodic windows. When parameter a varies in the range of [0.5, 1.1], system (2) is periodic and exhibits a period-doubling process; then, the system enters a chaotic state around a 1.1. After that system (2)  When a 2.4, c 1.7, d 2.5, the initial condition is (1, 0, 1) and b varies in (0, 1.5), the corresponding bifurcation and   Lyapunov exponents are shown in Figure 5. From Lyapunov exponents and bifurcation, we observe that the system undergoes a typical inverse-period-doubling process from chaos.  (2) gives cycle-4, cycle-2, and cycle-1 limit cycles.
To observe the memristive computation influence on system dynamics, here the coefficient d implying memristor resistance is used as a bifurcation parameter. When a 2.4, b 0.5, and c 1.7, and initial condition (1, 0, 1), and d increases in the region of [0, 5], the bifurcation and Lyapunov exponents are shown in Figures 6A,B    periodic doubling bifurcation happens when d becomes larger.
As shown in the latter, the newly introduced memristor does not break the property of offset boosting. Suppose if there is a constant e is introduced in the last dimension, Different constants of e revise the offset of the system variable x directly. As shown in Figure 7, three chaotic attractors with different offsets are displayed in cyan, green, and red. Here signal x(t) at e 0 is bipolar signal, while signals at x(t) at e ±3 are unipolar positive and negative signal. Offset controller e only rescales the average value of x without influencing system dynamics.

A MEMRISTIVE HINDMARSH-ROSE NEURON
The 2D HR neuro was simplified from the classical Hodgkin-Huxley model [25] by HR, which is expressed by the following expression, where a and b are positive constants, and I is the external current.
In the system (9), x stands for membrane potential, and y stands for spiking variable, [38]. To better express the complex dynamical behaviors of the neuron electrical activities, here a new 3-D memristive HR neuron model is proposed [39]. The external current I in the 2-D HR neuron model (8) be replaced by a memristor, the new HR neuron model can be described as, Let the left-hand side of the system (10) be equal to zero, the equilibria can be solved by the following equation: System (10) has a non-zero equilibrium point S (0.58771263, − 0.89973376, − and 0.58771263). Linearizing system (10) at the equilibrium point S 0 , the Jacobian matrix can be obtained as, The characteristic equation is derived as: det(λI − J) a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4 0 (13) where a 1 , a 2 , a 3 , and a 4 are polynomials containing a, b, c, d, and variables x and z: a 1 1, a 2 (−3x+1−2a)x−c+d|z|+1, a 3 −(x+1)(3x 2 +2ax−c+d|z|)−dxsgn(z)(2x+z) +(2b+1)x, and a 4 −x(3x 2 +2ax+2bx−dsgn(z)(2x+z)− c+d|z|). When The parameters in (10) are specified as follows: b 5.5, c 1.7, and d 2.5, and the initial condition is (0, 0, 0). When a varies within the range of [2, 3.3], the bifurcation diagram of the membrane potential x and Lyapunov exponents are shown in Figures 8A,B. When the parameter a is increased in the region [2, 2.6], system (10) exhibits typical period-doubling bifurcation; when the range of parameter a varies in [2.6, 2.92], system (10) shows chaos; When parameter a is in the range of [2.92, 3.3], system (10) exhibits an inverse-period-doubling process. As shown in Figure 9, the neuronal electrical activity shows different modes by modifying the value of the system parameter a. Corresponding attractors are displayed in Figure 10.
When a 2.8, c 1.7, and d 2.5, and the initial condition is (0, 0, 0), the bifurcation diagram of the maxima X of the membrane potential x and Lyapunov exponents of parameter b varies in the range of [2,13] are shown in Figures 11A,B. When parameter b is increased in the region of [0, 5.25], the system (10) exhibits typical period-doubling bifurcation phenomenon; When parameter b varies in [5.25, 5.77   Frontiers in Physics | www.frontiersin.org October 2021 | Volume 9 | Article 759913 6 period again. As shown in Figure 12, the bursting state of neuronal electrical activity can be changed into different modes by modifying the value of the system parameter b. As shown in Figure 13, attractors correspond to the different bursting states of neuronal electrical activity.
Also to observe the memristive computation influence on system dynamics, here the coefficient d is used as a bifurcation parameter in the derived HR neuron model. When a 2.4, b 5.5, c 1.7, and the initial condition is (0, 0, 0), the bifurcation diagram of the maxima X of the membrane potential x and  Lyapunov exponents of parameter d varies in the range of [1,4] are shown in Figures 14A,B. When the parameter d is increased in the region of [0, 2.24], The system exhibits typical period-doubling bifurcation phenomenon; When the range of the parameter d vary in [2.24, 2.75], the system (10) is chaotic, obviously, there are two typical period windows in this chaotic range; When parameter d is in the range of [2.75, 4], system (10) exhibits an inverse-period-doubling process. Interestingly, the function of the parameter d is similar to the system parameter a.
As shown in the latter, the newly introduced memristor does not break the property of offset boosting. Suppose there is a constant e introduced in both of the first dimension and the second dimension, Different constants of e revise the offset of the system variable y directly. As shown in Figure 15, three chaotic attractors with different offsets are displayed in cyan, green, and red. Here signal x(t) at e 0 is bipolar signal, while signals at x(t) at e ±3.5 are unipolar positive and negative signal. Offset controller e only rescales the average value of y without influencing system dynamics.

CONCLUSION
Memristor and memristive computation have great merits for producing chaos and dynamics control due to the special nonlinearity. It shows that even a memristor function is a linear function, the memory effect from memristive computation still returns chaos under a specific bifurcation. In this paper, two systems are reformed to be memristive chaotic systems based on the same memristor function. Offset boosting is discussed in both systems. Memristive computation as a new type of computing shows great potential with chaos-based engineering and pattern recognition in artificial intelligence, which deserves further research and will yield great value in information engineering.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

AUTHOR CONTRIBUTIONS
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.  Frontiers in Physics | www.frontiersin.org October 2021 | Volume 9 | Article 759913 8