Quite recently found by the electrical engineers Sara Dadras and Hamid Momeni in 2009, this attractor is special because it changes its scrollness when varying just one parameter.¹ At \(c = 4.7\), the chaotic attractor shows just two wings, at \(c = 3.9\) it suddenly evolves four scroll-like wings and at \(c = 1.7\) it goes back to tree scrolls. Around these values several things happen:

• \(0 < c \le 0.77\): single stable point
• \(0.77 < c \le 1.5\): single stable oscillation
• \(1.5 < c \le 4.45\): chaos
• \(4.45 < c < 5.17\): several regions of stable oscillations with chaotic outbreaks inbetween
• \(5.17 \le c < 7\): multiple stable limit cycles

Renders

Three-Scroll

Differential system:

\[\dot{x} = y - a\, x + b\, y\, z\] \[\dot{y} = z + c\, x - x\, z\] \[\dot{z} = d\, x\, y - h\, z\]

Constants:

\[a = 3\] \[b = 2.7\] \[c = 1.7\] \[d = 2\] \[h = 9\]

Four-Scroll

As above, \(c = 3.9\).

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1. S. Dadras and H. R. Momeni, 2009. "A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors". Phys. Let. A. 373(40). doi:10.1016/j.physleta.2009.07.088. ↩