Chuanna: Chua's Double Scroll
Mechanics, Chaos & Fluids arc, page 96 of 193. Back to The Pixelated Physics Universe
Who they are
Chuanna wires up five parts on her breadboard, and out spills a ribbon that loops around two points and never repeats.
Chuanna is a character of the Mechanics, Chaos & Fluids arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
Leon Chua designed a simple autonomous electronic circuit in 1983: two capacitors, an inductor, a resistor and one piecewise-linear nonlinear resistor, the Chua diode. It oscillates chaotically. The orbit loops around two unstable equilibria and jumps irregularly between them, tracing the “double scroll”. With the classic parameters a = 15.6, b = 28, m0 = −8/7 and m1 = −5/7, the dimensionless equations below give the double-scroll attractor.
$$ \begin{gathered} \frac{dx}{dt} = \alpha\,(y - x - f(x)), \qquad \frac{dy}{dt} = x - y + z, \qquad \frac{dz}{dt} = -\beta y \\ f(x) = m_1 x + \tfrac{1}{2}(m_0 - m_1)\left(|x+1| - |x-1|\right) \\ \alpha = 15.6, \quad \beta = 28, \quad m_0 = -\tfrac{8}{7}, \quad m_1 = -\tfrac{5}{7} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $x,\ y,\ z$ | scaled capacitor voltages and inductor current | 1 |
| $t$ | scaled time | 1 |
| $\alpha,\ \beta$ | circuit parameters (printed a, b on the shirt) | 1 |
| $m_0,\ m_1$ | inner and outer slopes of the Chua diode (printed m0, m1) | 1 |
Honest caveat: The Chua diode is itself built from op-amps or diodes in practice, so “five parts” counts it as one. Chaos was proved in 1986, not 1983.
Sources
- Matsumoto (1984) A chaotic attractor from Chua's circuit, IEEE Trans. Circuits Syst. 31, 1055
- Chua, Komuro & Matsumoto (1986) The double scroll family, IEEE Trans. Circuits Syst. 33, 1072
- Scholarpedia: Chua circuit
Crossovers
Chuanna meets Arnold's Cat (Cat Map Chaos). Chuanna's circuit and Arnold's Cat are both chaotic, but in different ways. The cat is a map that preserves area; Chuanna's flow is dissipative and settles onto a strange attractor.
Chuanna meets Flickermoth (1/f Pink Noise). Chuanna's double scroll never repeats, and its spectrum is broadband. Flickermoth's 1/f noise is a different kind of broadband signal; it shows up in many systems, and there isn't one cause for all of them.
Wear the lore
- Chuanna – Chua's Double Scroll Physics T-Shirt
- Related infographic poster: The Rössler Band – Plate 084 Physics Poster
Concept keywords: Chua's Double Scroll, Chuanna, Mechanics, Chaos & Fluids, physics lore, Pixelated Physics