Dynamics of Chaotic 3D Attractors: Part 2

This is a sequel to my repository Dynamics of Chaotic 3D Attractors: Part 1 on which I have reached the limit for using mathematical expressions. This repository will also only consider 3D attractors.

The plots are also available on Pinterest and Behance:

Relevant Repositories:

• Dynamics of Chaotic 3D Attractors: Part 1
• Dynamics Of Hyperchaotic High-Dimensional Attractors

The Sambas—Benkouider—Kaçar Attractor

Reference:
Sambas, A., Benkouider, K., Kaçar, S., Ceylan, N., Vaidyanathan, S., Sulaiman, I. M., Mohamed, M. A., Ayob, A. F. M., & Muni, S. S. (2024). Dynamic Analysis and Circuit Design of a New 3D Highly Chaotic System and its Application to Pseudo Random Number Generator (PRNG) and Image Encryption. SN Computer Science, 5(4).

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = \alpha\left(y-x\right) + yz, \\ \frac{\mathrm{d}y}{\mathrm{d}t} = x\left(\beta-z\right)-1, \\ \frac{\mathrm{d}z}{\mathrm{d}t} = x^2+xz-\varsigma z, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma \end{bmatrix} = \begin{bmatrix} 20\\ 50\\ 10 \end{bmatrix}. $$

The Xu—Wang Attractor

Reference:
Xu, Y., & Wang, Y. (2014). A new chaotic system without linear term and its impulsive synchronization. Optik - International Journal for Light and Electron Optics, 125(11), 2526–2530.

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = \ln\left(\alpha+\mathrm{e}^{y-x}\right) \\ \frac{\mathrm{d}y}{\mathrm{d}t} = xz, \\ \frac{\mathrm{d}z}{\mathrm{d}t} = \beta -xy, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta \end{bmatrix} = \begin{bmatrix} 0.1\\ 0.25 \end{bmatrix}. $$

The Sanum—Srisuchinwong Attractor

Reference:
Sanum, W., & Srisuchinwong, B. (2012). Highly Complex Chaotic System with Piecewise Linear Nonlinearity and Compound Structures. Journal of Computers, 7(4).

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = y-x \\ \frac{\mathrm{d}y}{\mathrm{d}t} = -z\tanh x, \\ \frac{\mathrm{d}z}{\mathrm{d}t} = -\alpha + xy +|y|, \end{cases} $$

$$ \alpha = 60. $$

The Zhou—Wang—Wu Attractor

Reference:
Zhou, W., Wang, Z., Wu, M., Zheng, W., & Weng, J. (2015). Dynamics analysis and circuit implementation of a new three-dimensional chaotic system. Optik - International Journal for Light and Electron Optics, 126(7-8), 765–768.

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = \alpha y \\ \frac{\mathrm{d}y}{\mathrm{d}t} = -z\text{sgn}\left(x\right) - \varsigma y, \\ \frac{\mathrm{d}z}{\mathrm{d}t} = \beta x^2 -1, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma \end{bmatrix} = \begin{bmatrix} 14\\ 3\\ 1 \end{bmatrix}. $$

The Kingni—Pham—Jafari Attractor

Reference:
Kingni, S. T., Pham, V.-T., Jafari, S., Kol, G. R., & Woafo, P. (2016). Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form. Circuits, Systems, and Signal Processing, 35(6), 1933–1948.

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = z \\ \frac{\mathrm{d}y}{\mathrm{d}t} = z^3 + z^2 + 3xz, \\ \frac{\mathrm{d}z}{\mathrm{d}t} = x^2+y^2-a^2-4yz^2, \end{cases} $$

$$ \alpha=0.991. $$

The Vaidyanathan Attractor

Reference:
Vaidyanathan, S. (2015). Analysis, properties and control of an eight-term 3-D chaotic system with an exponential nonlinearity. International Journal of Modelling, Identification and Control, 23(2), 164.

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = \alpha\left(y-x\right) + yz \\ \frac{\mathrm{d}y}{\mathrm{d}t} = \beta x - \varsigma xz, \\ \frac{\mathrm{d}z}{\mathrm{d}t} = \mathrm{e}^{xy}-\delta z+x^2, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma\\ \delta \end{bmatrix} = \begin{bmatrix} 11\\ 40\\ 0.4\\ 9 \end{bmatrix}. $$

The Zhang—Han Attractor

Reference:
Zhang, M., & Han, Q. (2016). Dynamic analysis of an autonomous chaotic system with cubic nonlinearity. Optik, 127(10), 4315–4319.

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = -\alpha x + \beta yz \\ \frac{\mathrm{d}y}{\mathrm{d}t} = -\varsigma y ^3 + \delta xz, \\ \frac{\mathrm{d}z}{\mathrm{d}t} =\varepsilon z - \vartheta xy, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma\\ \delta\\ \varepsilon\\ \vartheta \end{bmatrix} = \begin{bmatrix} 2\\ 10\\ 6\\ 3\\ 3\\ 1 \end{bmatrix}. $$

The Tuna—Koyuncu—Fidan Attractor

Reference:
Tuna, M., Koyuncu, I., Fidan, C. B., & Pehlivan, I. (2015). Real time implementation of a novel chaotic generator on FPGA. 2015 23nd Signal Processing and Communications Applications Conference (SIU).

$$ \begin{cases} \frac{\mathrm{d}x}{\mathrm{d}t} = y\left(z-\alpha\right) \\ \frac{\mathrm{d}y}{\mathrm{d}t} = y\left(z-\alpha\right)-x\left(z+\alpha\right), \\ \frac{\mathrm{d}z}{\mathrm{d}t} =-y\left(\alpha x - y\right)-\beta\left(z-\alpha\right), \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta \end{bmatrix} = \begin{bmatrix} 1.3\\ 4 \end{bmatrix}. $$