Chua’s Circuit Double Scroll

Chaos you can build on a bench

Many chaotic systems are abstract equations, but this one is a real circuit you can solder together. Leon Chua designed it in 1983 to be as simple as possible while still being chaotic. It has two capacitors, an inductor, a resistor and one special part, a nonlinear resistor now called Chua’s diode, whose current does not rise in proportion to the voltage across it but bends in two places.

Written in dimensionless form, with x and y the voltages across the two capacitors and z the current through the inductor, the circuit obeys:

dx/dt = α(y − x − g(x)), dy/dt = x − y + z, dz/dt = −β·y,

where g(x) = m₁x + ½(m₀ − m₁)(|x + 1| − |x − 1|) is the diode’s bent line, with slopes m₀ = −8/7 in the middle and m₁ = −5/7 outside. The negative slopes mean the diode feeds energy into the circuit instead of soaking it up.

Two scrolls

The circuit has two unstable resting states, one on each side. Near each one, the voltages spiral outwards in a widening scroll. When a spiral grows big enough, the voltage crosses into the other half of the diode’s characteristic and the motion is captured by the other scroll. When it jumps, and how many turns it makes on each side, is chaotic: the sequence never repeats. The “Jumps between scrolls” readout counts these crossings.

Turning the knob

α depends on the ratio of the two capacitors, and β on the inductor and resistor. With the classic values α = 15.6 and β = 28 you get the full double scroll. At α = 14 the motion stays on one side, a single scroll that is still chaotic. At α = 12.5 it settles into a simple repeating loop round one resting state. We checked each preset numerically, measuring the chaos rate λ each time.

On the oscilloscope

Engineers usually see this attractor by connecting the two capacitor voltages to an oscilloscope’s horizontal and vertical inputs, so the screen traces y against x, with the vertical gain turned up because y swings much less than x. This view mimics that: a dark glass screen, a graticule grid and green phosphor-coloured traces, with brighter colour for faster changes. Real circuits run at audio frequencies or faster, so the real pattern would fill the screen in a fraction of a second.

For the curious

Particles start on the attractor and are advanced with fourth-order Runge–Kutta in fixed steps of 0.005 dimensionless time units. Because the diode’s characteristic is made of straight pieces, the system is linear inside each region, which is part of what made its chaos possible to prove mathematically.