A simple memristive jerk system

School of Artificial Intelligence, Nanjing University of Information Science & Technology, Nanjing, China Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET), Nanjing University of Information Science & Technology, Nanjing, China College of Letters and Science, University of Wisconsin–Madison, Madison, Wisconsin, USA Department of Electrical and Computer Engineering, University of Michigan, Ann Arbor, Michigan, USA

Jerk systems are a simple type of dynamical system that can generate chaos [16][17][18][19][20]. Such a compact structure is composed of a couple integral operation units in series. Some jerk systems are chaotic when they contain a nonlinearity from quadratic terms [16][17][18], an exponential function [19] or a cubic term [20]. All other jerk systems that produce chaos based on the memristor are 4-D [21,22]. The novelty of this work is that compared with prior studies, we aim to construct a completely 3-D jerk memristive circuit. The most important challenge in this work is introducing a suitable memristor to break the existing oscillation in a 2-D structure and finally bring chaos.
In addition, many memristive systems exhibit chaotic oscillation combined with other states because of the integral effect from the memristor [6][7][8][9][10][21][22][23][24][25]. Unlike these memristive jerk systems, however, a second-order jerk structure with a memristor is explored that is both simple and robust for giving chaos. In Section 2, the model is given and analyzed with basic dynamical analysis. In Section 3, the circuit is built for proving the theoretical analysis. Finally, a short conclusion is made to summarize the work.

| SYSTEM MODEL
A simple chaotic jerk oscillator containing a memristor as one of the state variables was found by an exhaustive computer search based on the Euler method. Suppose there is a 2-D jerk structure _ y ¼ z; _ z ¼ f ðy; zÞ, and introduce a flux-controlled memductance W(x) in it. The following system is found for producing chaos: where the flux-controlled memductance W(x) = 1.3x 2 -1 is introduced in the z-dot equation. Here the variables y and z are the external system variables, and x is the internal variable in the memristor and indicates the magnetic flux. When a = 0.239 and b = 1, System (1) produces chaos with Lyapunov exponents (0.0529, 0, −1.0529) after a time of t = 2e7 and a corresponding Kaplan-Yorke dimension of D KY = 2.0502 for initial condition (0, −2, −2), as shown in Figure 1, whose basins of attraction (in the z = 0 plane) are shown in Figure 2.
This system is fairly delicate with chaos in only a narrow range of the parameter space, as indicated in Figure 3. The internal variable x comes from the integration of the system state variable y. System (1) is asymmetric with a speed of volume contraction determined by a derivative proposed by Lie: In addition, System (1) has no equilibria [26][27][28][29][30], and therefore the attractor is hidden [31][32][33][34][35].
More interesting is that unlike other memristive systems [8][9][10], System (1) outputs relatively stable oscillations except when switching between chaos and sliding initial values that agree with the plots of the dynamical region and basins of attraction. To further verify this, the solution based on offset boosting under a fixed initial condition is used for diagnosing multi-stability [36,37]. Taking offset-boosting d in the dimension y → y þ d, it is shown that when offset d varies in [-5, 5] except for the long transient process, System (1) remains chaotic unless it is dragged sliding with the initial condition, as shown in Figure 4.
The embedded memristor is defined as Flux-dependent memductance is related to the internal variable x, which is of quadratic degree [13,38,39] and is determined by the system variable y:

| JERK CIRCUIT IMPLEMENTATION
Obtaining an analog circuit to realize System (1) is a relatively easy way to introduce a memristor into the operational amplifier-based integration circuit. First, we construct a 2-D jerk main structure. Second, an equivalent circuit is designed for the applied memristor without resorting to another amplifier-based integration element. The basic principle is based on the characteristics of virtual break and virtual short of an operational amplifier. An analog circuit based on Equation (1) is designed as shown in Figure 6, and according to the Kirchhoff law, the circuit equations can be written as follows: where the memristor W(x) is equivalent to the circuit simulator as shown in Figure 7, In this process, a 2-D jerk structure provides a simple structure for the proposed system. The circuit structure contains two parts based on the integration including addition and subtraction of the system variables y and z according to Equation (1). Here the analog multiplier AD633/AD is applied to realize the nonlinear product operation, while the operational amplifier OPA404/BB combined with its peripheral circuit is used to construct the addition, inversion, and integration operations. The memductance W ðxÞ ¼ 1:3x 2 − 1 is realized through the circuit parameters C a = 100 nF, R a = 10 kΩ, R b = 7.69 kΩ, and R c = 10 kΩ. The system parameters a = 0.239, b = 1 are realized through the circuit parameters R 1 = R 3 = 10 kΩ, R 2 = 41.8 kΩ, R 0 = 30 kΩ, and V b = 3V. The capacitor C 1 = C 2 = 100 nF are selected for establishing a robust phase trajectory that only rescales the time of the oscillation. The equations used in the circuit implementation contain different constants from those in System (1) owing to amplitude and time rescaling, which is normal in circuit implementation. Figure 8 displays a plot of the experimental constraints of memductance and inherent relation of pinched hysteresis, while Figure 9 displays the phase portraits observed in the oscilloscope.
The experimental constraints of memductance, the inherent pinched hysteresis effect, and the experimental phase portraits agree well with the numerical simulation, proving the system dynamics and the effectiveness of the hardware circuit. As a main element, the equivalent memristor can potentially have a great effect on the performance of the jerk system, which is dominantly determined by two analog multipliers and one operational amplifier. These three components define the memristor applied in this work, and therefore some other physical memristor models (e.g. the HP memristor) cannot guarantee chaos in the 3-D jerk structure.

| CONCLUSIONS
By introducing a memristor into a second-order jerk structure, chaotic oscillation is found in a 3-D jerk system. The proposed simple memristive jerk system has only six terms while without any equilibria, one of which is quadratic. Circuit experiments show the same oscillation, and thus they agree with the numerical simulation. When flux-controlled memductance is revised as W ðxÞ ¼ 1:3 | x | −1, a minor parameter adjustment (a = 0.432, b = 1) can still recover chaos with Lyapunov exponents (0.0328, 0, −1.0332) and a corresponding attractor dimension of D KY = 2.0321, which simplifies the circuit realization. Compared with other memristive systems [10,40], this system is also unique for its robust chaotic oscillation, although it is hidden [41,42]. This feature is attractive for its application in chaos-based communication or image encryption. Future work on this circuit can investigate the introduction of other memristors such as the HP memristor into this proposed jerk structure for chaos. -391