Thomas Aizawa Lorenz Dadras Chen Lorenz84 Rössler Halvorsen Rabinovich-Fabrikant Three-Scroll Unified Sprott Wang-Suu

Strange attractors in 3D


Thomas

Thomas attractor

Parameter:

$b=0.208186$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&\sin y -bx\\ \frac{dy}{dt}&=&\sin z -by\\ \frac{dz}{dt}&=&\sin x-bz \end{eqnarray*}\right. $$

Aizawa

Aizawa attractor

Parameters:

$a = 0.95,$ $b = 0.7,$ $c = 0.6,$
$d = 3.5,$ $e = 0.25,$ $f = 0.1$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&(z - b) x - d y\\ \frac{dy}{dt}&=& d x + (z - b) y\\ \frac{dz}{dt}&=&c + a z - \frac{z^3}{3} - (x^2+y^2)(1+e z)+f z x^3 \end{eqnarray*}\right. $$

Lorenz

Lorenz attractor

Parameters:

$p = 10,$ $r = 28,$ $b = 8/3$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&p (-x + y)\\ \frac{dy}{dt}&=& -x z + r x - y \\ \frac{dz}{dt}&=& x y - b z \end{eqnarray*}\right. $$

Dadras

Dadras attractor

Parameters:

$a = 3,$ $b = 2.7,$ $c = 1.7,$
$d = 2,$ $e = 9$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=& y - a x +b y z\\ \frac{dy}{dt}&=& c y - x z +z\\ \frac{dz}{dt}&=& d x y - e z \end{eqnarray*}\right. $$

Chen

Chen attractor

Parameters:

$\alpha = 5$, $\beta = -10$, $\delta = -0.38$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&\alpha x- y z\\ \frac{dy}{dt}&=&\beta y + x z \\ \frac{dz}{dt}&=&\delta z + x y/3 \end{eqnarray*}\right. $$

Loren84

Lorenz84 attractor

Parameters:

$a = 0.95,$ $b = 7.91,$ $f = 4.83,$ $g = 4.66$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&-a x - y^2 - z^2 + a f\\ \frac{dy}{dt}&=& -y + x y - b x z + g\\ \frac{dz}{dt}&=& -z + b x y + x z \end{eqnarray*}\right. $$

Chen

Rössler attractor

Parameters:

$a = 0.2$, $b = 0.2$, $c = 5.7$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&-(y+z)\\ \frac{dy}{dt}&=&x+ay\\ \frac{dz}{dt}&=&b+z(x-c) \end{eqnarray*}\right. $$

Loren84

Halvorsen attractor

Parameter:

$a = 1.89$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&-a x-4y-4z-y^2 \\ \frac{dy}{dt}&=& -a y-4z-4x-z^2 \\ \frac{dz}{dt}&=& -a z-4x-4y-x^2 \end{eqnarray*}\right. $$

Rabinovich-Fabrikant

Rabinovich-Fabrikant attractor

Parameters:

$\alpha = 0.14$, $\gamma = 0.10$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&y ( z - 1 + x^2 ) + \gamma x\\ \frac{dy}{dt}&=&x ( 3 z + 1 - x^2 ) + \gamma y\\ \frac{dz}{dt}&=& - 2 z ( \alpha + x y ) \end{eqnarray*}\right. $$

Tscus

Three-Scroll Unified Chaotic System attractor

Parameter:

$a = 32.48,$ $b = 45.84,$ $c = 1.18,$
$d = 0.13,$ $e = 0.57$, $f = 14.7$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=&a(y-x)+d x y \\ \frac{dy}{dt}&=& bx-x z + f y \\ \frac{dz}{dt}&=& cz+xy-ex^2 \end{eqnarray*}\right. $$

Sprott

Sprott attractor

Parameters:

$a = 2.07$, $b = 1.79$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=& y + a x y +x z \\ \frac{dy}{dt}&=& 1 - b x^2 +yz \\ \frac{dz}{dt}&=& x-x^2-y^2 \end{eqnarray*}\right. $$

Tscus

Wang-Su attractor

Parameter:

$a = 0.2,$ $b = 0.01,$ $c = -0.4$

System:

$$\left\{ \begin{eqnarray*} \frac{dx}{dt}&=& ax+yz \\ \frac{dy}{dt}&=& b x + cy - xz \\ \frac{dz}{dt}&=& -z-xy \end{eqnarray*}\right. $$


References

  1. Dadras, S., Momeni, H.R. (2009). A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Physics Letters A. Volume 373, Issue 40. pp. 3637-3642.
  2. Lorenz E. N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences. 20(2): 130–141.
  3. Pan, L., Zhou, W., Fang,J., Li, D. (2010). A new three-scroll unified chaotic system coined. International Journal of Nonlinear Science. vol. 10, 462-474.
  4. Rössler, O. E. (1976). An Equation for Continuous Chaos. Physics Letters, 57A (5): 397–398.
  5. Solís Pérez, J. E., Gómez-Aguilar, J. F., Baleanu, D., Tchier, F. (2018). Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors. Entropy. 2018, 20(5), 384.
  6. Sprott. J. C. (2014). A dynamical system with a strange attractor and invariant tori Physic Letters A, 378 1361-1363.
  7. Thomas, René. (1999). Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ‘labyrinth chaos’. Int. J. Bifurcation and Chaos. 9 (10): 1889–1905.
  8. Tam L., Chen J., Chen H., Tou W. (2008). Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation. Chaos, Solitons and Fractals. Vol. 38, 826-839.
  9. Wang, Z., Sun, Y., van Wyk, J. B, Qi, G, van Wyk, M. A. (2009). A 3-D four-wing attractor and its analysis. Brazilian Journal of Physics, vol. 39, no. 3. pp.547-553.

Notes

The code can be found here

For the 3d enviroment I used the p5.EasyCam library developed by Thomas Diewald. I used the dat.GUI library for the controls

Finally, the position of a particle is computed with a 4th order Runge-Kutta method.