Chua's Circuit: A Chaotic Attractor Built From a Resistor

The simplest circuit proven to be chaotic

By the early 1980s, chaos was well established in mathematics — the Lorenz equations, the logistic map — but every chaotic system with a real physical embodiment was either too complex to analyse cleanly or lacked a rigorous proof that it was actually chaotic rather than just complicated-looking. In 1983, Leon Chua set out to design the simplest possible electronic circuit that could be proven chaotic, and built it. Chua's circuit uses two capacitors, one inductor, one linear resistor, and one deliberately nonlinear element now called the Chua diode — five components in total.

The three equations

The circuit's state — the voltages across its two capacitors and the current through its inductor — is described by three coupled ordinary differential equations in the dimensionless variables x, y and z:

dx/dt = α (y - x - f(x))
dy/dt = x - y + z
dz/dt = -β y

f(x) = m1·x + 0.5·(m0 - m1)·(|x + 1| - |x - 1|)     ← the Chua diode

Everything nonlinear in the system is concentrated in that one function f(x), the Chua diode: a piecewise-linear negative-resistance element whose current-voltage curve is three straight-line segments instead of a smooth curve. It can be built from two op-amps and a handful of resistors, which is precisely why Chua's circuit is not just a set of equations on a page — it is a circuit that can be assembled on a breadboard and its chaos measured with an oscilloscope, unlike the Lorenz system, which was derived from fluid convection and has no simple direct electronic realisation.

The double scroll

The Chua diode's three linear segments carve state space into three regions, each governed by its own locally linear dynamics with its own equilibrium point. The circuit's most famous behaviour, the double-scroll attractor, happens because two of those equilibria are both unstable in a spiralling way — a trajectory winds outward around one of them, and once it strays far enough it gets thrown across to orbit the other one instead, then eventually crosses back. The result is two spiral "scrolls" joined by an irregular switching pattern between them, and because that switching is governed by the sensitive, chaotic core of the dynamics, no two orbits ever retrace exactly the same path.

Sensitive dependence, made visible

Chua's circuit has a positive Lyapunov exponent: start two trajectories at nearly identical initial states and the distance between them grows exponentially with time, even though the underlying equations are completely deterministic and involve no randomness at all. That exponential divergence — the same phenomenon popularised as the butterfly effect — is the mathematical signature of chaos, and it is exactly what a numerical integrator has to reproduce faithfully. An integrator with too coarse a time step or the wrong order of accuracy will not just be slightly off; it will diverge from the true trajectory at the same exponential rate as any two nearby real trajectories would, which is why chaotic systems like this one are normally integrated with fourth-order Runge-Kutta (RK4) rather than a simple Euler step.

Why one circuit changed the field

Because Chua's circuit is both mathematically simple enough to prove chaotic rigorously (Chua, Komuro and Matsumoto formally proved it in 1986) and physically simple enough to build with off-the-shelf parts, it became the standard testbed for experimental chaos theory: control of chaos, synchronisation of chaotic systems, and chaos-based secure communication were all first demonstrated or refined using variants of this exact circuit. It occupies a role in nonlinear dynamics roughly analogous to the fruit fly in genetics — not important because it does anything useful on its own, but because it is simple enough that everyone can study the same well-understood system and compare results directly.

Frequently asked questions

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Everything above runs in your browser — open Chua's Circuit and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.