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:
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).
$$
\begin{cases}
\dot{x} = \alpha\left(y-x\right) + yz, \\
\dot{y} = x\left(\beta-z\right)-1, \\
\dot{z} = x^2+xz-\varsigma z,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix} = \begin{bmatrix}
20\\
50\\
10
\end{bmatrix}.
$$
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.
$$
\begin{cases}
\dot{x} = \ln\left(\alpha+\mathrm{e}^{y-x}\right) \\
\dot{y} = xz, \\
\dot{z} = \beta -xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix} = \begin{bmatrix}
0.1\\
0.25
\end{bmatrix}.
$$
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).
$$
\begin{cases}
\dot{x} = y-x \\
\dot{y} = -z\tanh x, \\
\dot{z} = -\alpha + xy +|y|,
\end{cases}
$$
$$
\alpha = 60.
$$
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.
$$
\begin{cases}
\dot{x} = \alpha y \\
\dot{y} = -z\text{sgn}\left(x\right) - \varsigma y, \\
\dot{z} = \beta x^2 -1,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix} = \begin{bmatrix}
14\\
3\\
1
\end{bmatrix}.
$$
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.
$$
\begin{cases}
\dot{x} = z \\
\dot{y} = z^3 + z^2 + 3xz, \\
\dot{z} = x^2+y^2-a^2-4yz^2,
\end{cases}
$$
$$
\alpha=0.991.
$$
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.
$$
\begin{cases}
\dot{x} = \alpha\left(y-x\right) + yz \\
\dot{y} = \beta x - \varsigma xz, \\
\dot{z} = \mathrm{e}^{xy}-\delta z+x^2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix} = \begin{bmatrix}
11\\
40\\
0.4\\
9
\end{bmatrix}.
$$
Reference: Zhang, M., & Han, Q. (2016). Dynamic analysis of an autonomous chaotic system with cubic nonlinearity. Optik, 127(10), 4315–4319.
$$
\begin{cases}
\dot{x} = -\alpha x + \beta yz \\
\dot{y} = -\varsigma y ^3 + \delta xz, \\
\dot{z} =\varepsilon z - \vartheta xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon\\
\vartheta
\end{bmatrix} = \begin{bmatrix}
2\\
10\\
6\\
3\\
3\\
1
\end{bmatrix}.
$$
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).
$$
\begin{cases}
\dot{x} = y\left(z-\alpha\right) \\
\dot{y} = y\left(z-\alpha\right)-x\left(z+\alpha\right), \\
\dot{z} =-y\left(\alpha x - y\right)-\beta\left(z-\alpha\right),
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta
\end{bmatrix} = \begin{bmatrix}
1.3\\
4
\end{bmatrix}.
$$
The Li—Pehlivan—Sprott Attractor Reference: Li, C., Pehlivan, İ., & Sprott, J. C. (2015). Amplitude-phase control of a novel chaotic attractor. TURKISH JOURNAL OF ELECTRICAL ENGINEERING & COMPUTER SCIENCES, 24, 1–11.
$$
\begin{cases}
\dot{x} = x-yz, \\
\dot{y} =-\alpha y + xz, \\
\dot{z} = -\beta y - \varsigma z + xy,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix} = \begin{bmatrix}
2.5\\
3.75\\
1.125
\end{bmatrix}.
$$
