Dynamics of Hyperchaotic Attractors The repository will feature a gallery of high-dimensional hyperchaotic attractors plotted by me in MATLAB based on General Algorithm of The Explicit Runge—Kutta Method .
The plots are also available on Pinterest and Behance:
The Hyperchaotic Mobayen—Mostafaee Attractor Reference: Mobayen, S., Mostafaee, J., Alattas, K. A., Ke, M., Hsueh, Y., & Zhilenkov, A. (2024). A new hyperchaotic system: circuit realization, nonlinear analysis and synchronization control. Physica Scripta, 99(10), 105204.
$$
\begin{cases}
\dot{x}_1 = \alpha_1 x_6, \\
\dot{x}_2 = -\alpha_2 x_4 - x_1 x_3, \\
\dot{x}_3 = -\alpha_3 x_3 + x_1 x_2, \\
\dot{x}_4 = \alpha_4 x_5, \\
\dot{x}_5 = \alpha_5 x_2 - x_7, \\
\dot{x}_6 = -\alpha_6 x_1 - x_5, \\
\dot{x}_7 = -\alpha_7 x_6 - x_1.
\end{cases}
$$
$$\begin{bmatrix}
\alpha_1 \\
\alpha_2 \\
\alpha_3 \\
\alpha_4 \\
\alpha_5 \\
\alpha_6 \\
\alpha_7
\end{bmatrix}
=
\begin{bmatrix}
90 \\
12 \\
200 \\
6 \\
5.17 \\
43.2 \\
30
\end{bmatrix}.$$
The Hyperchaotic Pang—Liu Attractor Reference: Pang, S., & Liu, Y. (2011). A new hyperchaotic system from the Lü system and its control. Journal of Computational and Applied Mathematics, 235(8), 2775–2789.
$$
\begin{cases}
\dot{x}_1 = \alpha\left(x_2-x_1\right) \\
\dot{x}_2 = -x_1x_3+\varsigma x_2+x_4, \\
\dot{x}_3 = x_1x_2-\beta x_3, \\
\dot{x}_4 = -\delta x_1 -\varepsilon x_2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon
\end{bmatrix} = \begin{bmatrix}
36\\
3\\
20\\
2\\
2
\end{bmatrix}.
$$
The Hyperchaotic Rössler Attractor Reference: Rossler, O. E. (1979). An equation for hyperchaos. Physics Letters A, 71(2-3), 155–157.
$$
\begin{cases}
\dot{x}_1 =-x_2-x_3 \\
\dot{x}_2 = x_1+\alpha x_2+x_4, \\
\dot{x}_3 = \beta+x_1x_3, \\
\dot{x}_4 = -\varsigma x_3+\delta x_4,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta
\end{bmatrix} = \begin{bmatrix}
0.25\\
3\\
0.5\\
0.05
\end{bmatrix}.
$$
The Hyperchaotic Dadras—Momeni—Qi Attractor Reference: Dadras, S., Momeni, H. R., Qi, G., & Wang, Z. (2011). Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form. Nonlinear Dynamics, 67(2), 1161–1173.
$$
\begin{cases}
\dot{x}_1 = \alpha x_1 -x_2x_3+x_4, \\
\dot{x}_2 = x_1x_3 -\beta x_2, \\
\dot{x}_3 = x_1x_2-\varsigma x_3+x_1x_4, \\
\dot{x}_4 = -x_2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix} = \begin{bmatrix}
8\\
40\\
14.9
\end{bmatrix}.
$$
The Hyperchaotic Mathieu—Van der Pol Attractor Reference: Li, S.-Y., Huang, S.-C., Yang, C.-H., & Ge, Z.-M. (2012). Generating tri-chaos attractors with three positive Lyapunov exponents in new four order system via linear coupling. Nonlinear Dynamics, 69(3), 805–816.
$$
\begin{cases}
\dot{x}_1 = x_2, \\
\dot{x}_2 = -\left(\alpha + \beta x_3\right)x_1-\left(\alpha+\beta x_3\right)x_1^3-\varsigma x_2 +\delta x_3, \\
\dot{x}_3 = x_4, \\
\dot{x}_4 = -\varepsilon x_3 + \vartheta\left(1-x_3^2\right)x_4+\zeta x_1 ,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon\\
\vartheta\\
\zeta
\end{bmatrix} = \begin{bmatrix}
91.17\\
5.023\\
-0.001\\
91\\
87.001\\
0.018\\
9.5072
\end{bmatrix}.
$$
The Hyperchaotic Li—Sprott—Thio Attractor Reference: Li, C., Sprott, J. C., & Thio, W. (2014). Bistability in a hyperchaotic system with a line equilibrium. Journal of Experimental and Theoretical Physics, 118(3), 494–500.
$$
\begin{cases}
\dot{x}_1 = x_2-x_1x_3-x_2x_3+x_4, \\
\dot{x}_2 = \alpha x_1x_3, \\
\dot{x}_3 = x_2^2-\beta x_3^2, \\
\dot{x}_4 = -\varsigma x_2,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma
\end{bmatrix} = \begin{bmatrix}
5\\
0.28\\
0.05
\end{bmatrix}.
$$
The Hyperchaotic Yi—Xiao—Yu Attractor Reference: Yi, L., Xiao, W., Yu, W., & Wang, B. (2018). Dynamical analysis, circuit implementation and deep belief network control of new six-dimensional hyperchaotic system. Journal of Algorithms & Computational Technology, 174830181878864.
$$
\begin{cases}
\dot{x}_1 = \alpha (x_2 - x_1) + x_4, \\
\dot{x}_2 = \varsigma x_1 - x_2 - x_1 x_3 - x_5, \\
\dot{x}_3 = -\beta x_3 + x_1 x_2, \\
\dot{x}_4 = \delta x_4 - x_2 x_3, \\
\dot{x}_5 = \vartheta x_2, \\
\dot{x}_6 = -\varepsilon x_6 + x_3 x_4,
\end{cases}
$$
$$
\begin{bmatrix}
\alpha\\
\beta\\
\varsigma\\
\delta\\
\varepsilon\\
\vartheta
\end{bmatrix} =
\begin{bmatrix}
10\\
\dfrac{8}{3}\\
28\\
-1\\
10\\
3
\end{bmatrix}.
$$
