# A Novel Four-Dimensional Memristive Hyperchaotic Map Based on a Three-Dimensional Parabolic Chaotic Map with a Discrete Memristor

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A 4D Parabolic Memristive Hyperchaotic Map Model

#### 2.1. A 3D Parabolic Chaotic Map

^{*}, y

^{*}, z

^{*}). The fixed point satisfies

^{2}, Equation (4) has only zero solutions; that is, the map (3) has only one origin fixed point S

_{0}= (0, 0, 0). When b > 1, in addition to one zero solution, Equation (4) has two non-zero real solutions; i.e., the map (3) has three fixed points. They are shown below as

#### 2.2. A Discrete Memristor Model

#### 2.3. A 4D Parabolic Memristive Hyperchaotic Map

^{*}, y′

^{*}, z′

^{*}, w′

^{*}) be the fixed point of the 4D memristive hyperchaotic map, and the fixed point satisfies

_{1}| < 1, |λ

_{2}| < 1, |λ

_{3}| < 1, |λ

_{4}| < 1; otherwise, it is unstable. From Equation (20), it can be seen that λ

_{1}and λ

_{2}are always at the center of the unit circle, namely, inside the unit circle, λ

_{3}is always on the unit circle, and λ

_{4}depends on the initial state of the parameters b and c as well as the internal state q of the memristor. In other words, the parameters b and c and the memristor-related parameters can change the stability of the four-dimensional memristive map.

## 3. Dynamical Analyses

#### 3.1. Lyapunov Exponential Spectrum and Bifurcation

#### 3.2. Phase Diagram and Iterative Sequence

#### 3.3. Transient Chaos and State Transfer

#### 3.4. Multi Stability Analysis

## 4. Hardware Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The DM frequency-dependent hysteresis loops for the same amplitude condition. (

**a**) Input signal frequency 2 rad/s. (

**b**) Input signal frequency 3 rad/s. (

**c**) Input signal frequency 4 rad/s.

**Figure 4.**The DM amplitude-dependent hysteresis loops for the same frequency condition. (

**a**) Input signal amplitude of 1A. (

**b**) Input signal amplitude of 0.8A. (

**c**) Input signal amplitude of 0.5A.

**Figure 6.**LEs and bifurcations for 3D parabolic chaotic map depending on the parameter b. (

**a**) LEs. (

**b**) Bifurcation diagram. (

**c**) Local view of bifurcation diagram.

**Figure 7.**LEs and bifurcations for 3D parabolic chaotic map depending on the parameter c. (

**a**) LEs. (

**b**) Bifurcation. (

**c**) Local view of bifurcation.

**Figure 8.**LEs and bifurcations for 3D parabolic chaotic map depending on the parameter a. (

**a**) Bifurcation of states x. (

**b**) Bifurcation of states y. (

**c**) Bifurcation of states z (

**d**) LEs.

**Figure 9.**LEs and bifurcations for 4D memristive hyperchaotic map depending on the parameter b. (

**a**) LEs. (

**b**) Bifurcation. (

**c**) Local view of the bifurcation.

**Figure 10.**LEs and bifurcations for 4D memristive hyperchaotic map depending on the parameter c. (

**a**) LEs. (

**b**) Bifurcation. (

**c**) Local view of the bifurcation.

**Figure 11.**LEs and bifurcations for 4D memristive hyperchaotic map depending on the parameter a. (

**a**) LEs. (

**b**) Local view LEs between the intervals [0.2,0.4]. (

**c**) Local view LEs between the intervals [1.2,1.4]. (

**d**) Bifurcation of states x. (

**e**) Bifurcation of states y. (

**f**) Bifurcation of states z.

**Figure 12.**Phase diagrams for 4D memristive hyperchaotic map. (

**a**) b = 1.31, (

**b**) b = 1.34, (

**c**) b = 1.45, (

**d**) b = 1.48, (

**e**) b = 1.56, (

**f**) b = 1.68, and (

**g**) x–y plane at b = 1.68, (

**h**) z–x plane at b = 1.68, and (

**i**) y–z plane at b = 1.68.

**Figure 13.**Iterative sequences for 4D memristive hyperchaotic map. (

**a**) Iterative sequence of the limit cycle state. (

**b**) Iterative sequence of the chaotic state. (

**c**) Iterative sequence of the hyperchaotic state.

**Figure 14.**Iterative sequence and attractor diagram of transient chaotic phenomena with parameter b = 1.492. (

**a**) Iterative sequence of y at transient chaos. (

**b**) Chaotic sequence. (

**c**) Periodic sequence. (

**d**) Chaotic attractor. (

**e**) Periodic attractor.

**Figure 15.**Iterative sequences and phase diagrams of the state transition phenomenon with parameter b = 1.34. (

**a**) Iterative sequence of y at state transfer. (

**b**) Periodic sequence. (

**c**) Quasi-periodic sequence. (

**d**) Periodic chaotic attractor. (

**e**) Quasi-periodic attractor.

**Figure 16.**Iterative sequences and phase diagrams of the state transition phenomenon with parameter b = 1.4985. (

**a**) Iterative sequence of y at state transfer. (

**b**) Chaotic sequence. (

**c**) Periodic sequence. (

**d**) Chaotic attractor. (

**e**) Periodic attractor.

**Figure 17.**Iterative sequences and phase diagrams of the state transition phenomenon with parameter c = 1.907. (

**a**) Iterative sequence of y at state transfer. (

**b**) Chaotic sequence. (

**c**) Periodic sequence. (

**d**) Chaotic attractor. (

**e**) Periodic attractor.

**Figure 18.**Dynamic behavior of parameters b with different initial values, (

**a**) LEs with initial values (0.6, 0.2, 0.5, 0.3), (

**b**) bifurcation diagram with initial values (0.6, 0.2, 0.5, 0.3), (

**c**) LEs with initial values (−0.6, 0.2, 0.5, −0.3).

**Figure 19.**Dynamic behavior of parameters c with different initial values. (

**a**) LEs with initial values (0.6, 0.2, 0.5, 0.3), (

**b**) bifurcation diagram with initial values (0.6, 0.2, 0.5, 0.3), (

**c**) LEs with initial values (−0.6, 0.2, 0.5, −0.3).

**Figure 20.**Typical coexistence phase diagrams for the memristive hyperchaotic map having various parameters b and c. (

**a**) b = 1.492, (

**b**) b = 1.6355, (

**c**) b = 1.6875, (

**d**) c = 1.7675, (

**e**) c = 1.907, (

**f**) c = 1.9672.

**Figure 22.**Oscilloscope-captured 2D planar phase diagrams of a 4D memristive map. (

**a**) x–y, (

**b**) z–x, (

**c**) y–z.

**Figure 23.**Oscilloscope-captured iterative sequences of different states of a 4D memristive map. (

**a**) limit cycle state, (

**b**) chaotic state, (

**c**) hyperchaotic state.

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## Share and Cite

**MDPI and ACS Style**

Wang, M.; Tong, L.; Li, C.; Zhang, X.; Iu, H.H.-C.; Li, Z.
A Novel Four-Dimensional Memristive Hyperchaotic Map Based on a Three-Dimensional Parabolic Chaotic Map with a Discrete Memristor. *Symmetry* **2023**, *15*, 1879.
https://doi.org/10.3390/sym15101879

**AMA Style**

Wang M, Tong L, Li C, Zhang X, Iu HH-C, Li Z.
A Novel Four-Dimensional Memristive Hyperchaotic Map Based on a Three-Dimensional Parabolic Chaotic Map with a Discrete Memristor. *Symmetry*. 2023; 15(10):1879.
https://doi.org/10.3390/sym15101879

**Chicago/Turabian Style**

Wang, Mengjiao, Luyao Tong, Chunlai Li, Xinan Zhang, Herbert Ho-Ching Iu, and Zhijun Li.
2023. "A Novel Four-Dimensional Memristive Hyperchaotic Map Based on a Three-Dimensional Parabolic Chaotic Map with a Discrete Memristor" *Symmetry* 15, no. 10: 1879.
https://doi.org/10.3390/sym15101879