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:

Relevant Repositories:

• Dynamics of Chaotic 3D Attractors: Part 1
• Dynamics of Chaotic 3D Attractors: Part 2

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.

$$ \begin{cases} \frac{\mathrm{d}x_1}{\mathrm{d}t} = \alpha\left(x_2-x_1\right) \\ \frac{\mathrm{d}x_2}{\mathrm{d}t} = -x_1x_3+\varsigma x_2+x_4, \\ \frac{\mathrm{d}x_3}{\mathrm{d}t} = x_1x_2-\beta x_3, \\ \frac{\mathrm{d}x_4}{\mathrm{d}t} = -\delta x_1 -\varepsilon x_2, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma\\ \delta\\ \varepsilon \end{bmatrix} = \begin{bmatrix} 36\\ 3\\ 20\\ 2\\ 2 \end{bmatrix}. $$

The Hyperchaotic Rössler Attractor

Reference:
Rossler, O. E. (1979). An equation for hyperchaos. Physics Letters A, 71(2-3), 155–157.

$$ \begin{cases} \frac{\mathrm{d}x_1}{\mathrm{d}t} =-x_2-x_3 \\ \frac{\mathrm{d}x_2}{\mathrm{d}t} = x_1+\alpha x_2+x_4, \\ \frac{\mathrm{d}x_3}{\mathrm{d}t} = \beta+x_1x_3, \\ \frac{\mathrm{d}x_4}{\mathrm{d}t} = -\varsigma x_3+\delta x_4, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma\\ \delta \end{bmatrix} = \begin{bmatrix} 0.25\\ 3\\ 0.5\\ 0.05 \end{bmatrix}. $$

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.

$$ \begin{cases} \frac{\mathrm{d}x_1}{\mathrm{d}t} = \alpha x_1 -x_2x_3+x_4, \\ \frac{\mathrm{d}x_2}{\mathrm{d}t} = x_1x_3 -\beta x_2, \\ \frac{\mathrm{d}x_3}{\mathrm{d}t} = x_1x_2-\varsigma x_3+x_1x_4, \\ \frac{\mathrm{d}x_4}{\mathrm{d}t} = -x_2, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma \end{bmatrix} = \begin{bmatrix} 8\\ 40\\ 14.9 \end{bmatrix}. $$

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.

$$ \begin{cases} \frac{\mathrm{d}x_1}{\mathrm{d}t} = x_2, \\ \frac{\mathrm{d}x_2}{\mathrm{d}t} = -\left(\alpha + \beta x_3\right)x_1-\left(\alpha+\beta x_3\right)x_1^3-\varsigma x_2 +\delta x_3, \\ \frac{\mathrm{d}x_3}{\mathrm{d}t} = x_4, \\ \frac{\mathrm{d}x_4}{\mathrm{d}t} = -\varepsilon x_3 + \vartheta\left(1-x_3^2\right)x_4+\zeta x_1 , \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma\\ \delta\\ \varepsilon\\ \vartheta\\ \zeta \end{bmatrix} = \begin{bmatrix} 91.17\\ 5.023\\ -0.001\\ 91\\ 87.001\\ 0.018\\ 9.5072 \end{bmatrix}. $$

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.

$$ \begin{cases} \frac{\mathrm{d}x_1}{\mathrm{d}t} = x_2-x_1x_3-x_2x_3+x_4, \\ \frac{\mathrm{d}x_2}{\mathrm{d}t} = \alpha x_1x_3, \\ \frac{\mathrm{d}x_3}{\mathrm{d}t} = x_2^2-\beta x_3^2, \\ \frac{\mathrm{d}x_4}{\mathrm{d}t} = -\varsigma x_2, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma \end{bmatrix} = \begin{bmatrix} 5\\ 0.28\\ 0.05 \end{bmatrix}. $$

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.

$$ \begin{cases} \frac{\mathrm{d}x_1}{\mathrm{d}t} = \alpha (x_2 - x_1) + x_4, \\ \frac{\mathrm{d}x_2}{\mathrm{d}t} = \varsigma x_1 - x_2 - x_1 x_3 - x_5, \\ \frac{\mathrm{d}x_3}{\mathrm{d}t} = -\beta x_3 + x_1 x_2, \\ \frac{\mathrm{d}x_4}{\mathrm{d}t} = \delta x_4 - x_2 x_3, \\ \frac{\mathrm{d}x_5}{\mathrm{d}t} = \vartheta x_2, \\ \frac{\mathrm{d}x_6}{\mathrm{d}t} = -\varepsilon x_6 + x_3 x_4, \end{cases} $$

$$ \begin{bmatrix} \alpha\\ \beta\\ \varsigma\\ \delta\\ \varepsilon\\ \vartheta \end{bmatrix} = \begin{bmatrix} 10\\ \frac{8}{3}\\ 28\\ -1\\ 10\\ 3 \end{bmatrix}. $$