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

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

Concept keywords: Chua's Double Scroll, Chuanna, Mechanics, Chaos & Fluids, physics lore, Pixelated Physics