Memristor is a new nonlinear component that brings great convivence for chaos generation [1–5] and dynamics control [6–10]. When a memristor is introduced, a dynamical system finds its way to exhibit more complicated evolvements. Whatever in a 3D [11–13] or 4D [14–16] or an even higher dimensional [17, 18] system, a memristor does endow abundant dynamics including hidden attractor [19, 20], bistability [21, 22], tristability [23], extreme multistability [24–27], and megamultistability [28–30]. Memristor, meminductor [31, 32], and memcapacitor [33, 34] are in fact representations of special computation, where the current output is critically associated with the past states and the current input. This is essentially a kind of time delay, which brings nonlinear factors and makes the system more prone to be chaos or even hyperchaos [35–37].

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.

The 2D HR neuro was simplified from the classical Hodgkin–Huxley model [25] by HR, which is expressed by the following expression, { x ˙ = y − x 3 + a x 2 + I , y ˙ = 1 − b x 2 − y . (9) 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, { x ˙ = y − x 3 + a x 2 − ( − c + d | z | ) x , y ˙ = 1 − b x 2 − y , z ˙ = − x 2 − x z . (10)

Let the left-hand side of the system (10) be equal to zero, the equilibria can be solved by the following equation: { y − x 3 + a x 2 − ( − c + d | z | ) x = 0 1 − b x 2 − y = 0 − x 2 − x z = 0 (11)

System (10) has a non-zero equilibrium point S = (0.58771263, − 0.89973376, − and 0.58771263). Linearizing system (10) at the equilibrium point S₀, the Jacobian matrix can be obtained as, J =   [ 3 x 2 + 2 a x − ( − c + d | z | ) 1 − d x s g n ( z ) - 2 b x − 1 0 − 2 x − z 0 − x ] (12)

The characteristic equation is derived as: det ( λ I - 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 = ( − 3 x + 1 − 2 a ) x − c + d | z | + 1 , a 3 = − ( x + 1 ) ( 3 x 2 + 2 a x − c + d | z | ) − d x sgn ( z ) ( 2 x + z ) + ( 2 b + 1 ) x , and a 4 = − x ( 3 x 2 + 2 a x + 2 b x − d sgn ( z ) ( 2 x + z ) − c + d | z | ) . When a =2.8, b = 5.5, c = 1.7, and d = 2.5, the eigenvalues of S are: λ 1 = 0.683273 , and λ 2,3 = 0.107353 ± 0.597675 i , thus showing it is a saddle point.

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], [6.68, 7.59], [8.5, 8.63], [8.89, 9.28], [10.84, 10.97], and [12.27, 12.4], system (4) exhibits chaos; When parameter b is in the range of [5.77, 6.68], [7.59, 8.5], [8.63 8.89], [9.28, 10.84], and [10.97, 12.27], system (10) gives periodic solution. When b increases in the region of [12.27, 13], system (10) moves from chaos to 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, { x ˙ = y − e − x 3 + a x 2 − ( − c + d | z | ) x , y ˙ = 1 − b x 2 − ( y − e ) , z ˙ = − x 2 − x z . (14)

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.

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.