🔌 Chua's Circuit

As the resistance parameter α is increased from a small value, the circuit first settles into a stable fixed point, then bifurcates to a limit cycle (period-1 oscillation). Increasing α further causes the period to double: period-2, period-4, period-8, … in a sequence that converges geometrically at the Feigenbaum ratio δ ≈ 4.669. After infinitely many doublings the period becomes infinite — chaos emerges. This universal period-doubling route to chaos was discovered by Feigenbaum in 1978 and applies to a vast class of nonlinear systems.

Chua's diode is a nonlinear resistor whose current–voltage characteristic is piecewise linear: I = m₁V + (m₀ – m₁)/2 · (|V + 1| – |V – 1|). It has a negative resistance slope in the inner region (|V| < 1), meaning it supplies energy rather than dissipating it, which destabilises the circuit. In hardware it is typically built from two op-amps and a few resistors forming a negative-impedance converter (NIC). Chua designed it specifically to be the simplest possible nonlinearity that produces chaos.

In the normalised Chua equations, α = C₂/(RC₁) is proportional to the ratio of the two capacitances and inversely proportional to the resistance R. β = C₂/(L G²) relates to the inductance L and conductance G. Together they set the time scales of the two RC sub-circuits relative to the LC tank circuit. Increasing α tends to drive the system into chaos; changing β shifts the frequency of the limit-cycle oscillation and the shape of the attractor.

Before Chua's circuit, chaos had been studied mainly in abstract mathematical models (Lorenz equations, logistic map) or in hard-to-reproduce fluid experiments. Chua's circuit gave experimentalists a simple, reproducible, inexpensive bench circuit — a few pounds of components — that reliably produces chaos, period-doubling bifurcations, and strange attractors. It has been used as a proof-of-concept for chaos-based secure communications (masking a message signal behind a chaotic carrier) and as a testbed for chaos control algorithms.

At certain parameter values the trajectory winds around only one of the two unstable equilibria (single scroll), while at others it switches between both (double scroll). The transition depends on whether the trajectory has enough energy to "jump" from one basin to the other. In the double-scroll regime the system explores both lobes of the attractor, producing the striking figure-eight shape in the X–Y phase plane and a broader, noisier power spectrum that is characteristic of chaos.

Yes — chaos synchronisation (Pecora and Carroll, 1990) showed that a slave Chua circuit driven by the output of a master circuit locks onto the same trajectory, even if an eavesdropper cannot predict it without knowing the exact initial conditions. A message signal can be added to the chaotic carrier at the transmitter and recovered at the synchronised receiver. While not cryptographically secure by modern standards (the chaotic signal can be broken with time-series analysis), it demonstrated an entirely new approach to analogue encryption.

Enabling this mode initialises a second trajectory with a perturbation of Δ = 10⁻⁵ from the first trajectory's current state. Both are integrated with the same parameters and drawn in different colours. In a periodic (non-chaotic) regime the two lines quickly overlap and remain together. In the chaotic regime they diverge exponentially and after a few dozen time units are completely uncorrelated, dramatically illustrating sensitive dependence on initial conditions — the "butterfly effect" in an actual electronic circuit.

The four presets trace the bifurcation sequence: "Limit cycle" sits at low α where the system settles into a single closed orbit; "Period-2" is a first period-doubling where the orbit visits two loops per cycle; "Double scroll" is in the full chaotic regime with λ > 0; "Single scroll" is an intermediate state with chaos confined to one lobe of the attractor. By cycling through these presets (or using the "Cycle α" button) you can observe the entire route to chaos in a few seconds.