Chua's Circuit Attractor

Overview

In 1983, professor Leon O. Chua at UC Berkeley proved that deterministic chaos was not just an abstract mathematical curiosity: it could be physically constructed using standard, off-the-shelf electronic components.

Chua’s circuit consists of four standard linear elements (two capacitors, an inductor, and a resistor) connected to a single active non-linear resistor, known universally today as Chua’s diode. The circuit generates electrical voltages and currents that fluctuate in an endless, non-repeating double-scroll strange attractor.

The Mathematical Model

The state variables $(x, y, z)$ correspond to the voltage across capacitor $C_1$, the voltage across capacitor $C_2$, and the electric current through the inductor $L$:

$\frac{dx}{dt} = \alpha (y - x - f(x))$

$\frac{dy}{dt} = x - y + z$

$\frac{dz}{dt} = -\beta y$

The piecewise-linear function $f(x)$ models the non-linear negative resistance of Chua’s diode:

$f(x) = m_1 x + \frac{m_0 - m_1}{2} \left(|x + 1| - |x - 1|\right)$

The Double-Scroll Phenomenon

• Bistable Spirals: The system possesses three equilibrium points: an unstable saddle at the origin and two symmetric equilibrium points on either side.
• The Double Scroll: Trajectories spiral outwards around one of the equilibrium points until reaching a critical threshold, where they are swept across the origin and captured by the opposite spiral.
• Experimental Verification: Chua’s attractor was the first chaotic system to be rigorously verified mathematically, simulated digitally, and observed in physical laboratory hardware on an oscilloscope.