Chua's circuit

Chua's circuit (also known as a Chua circuit) is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never "repeats". It was invented in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system, leading some to declare it "a paradigm for chaos". An autonomous circuit made from standard components (resistors, capacitors, inductors) must satisfy three criteria before it can display chaotic behaviour. It must contain: one or more nonlinear elements, one or more locally active resistors, three or more energy-storage elements. Chua's circuit is the simplest electronic circuit meeting these criteria. As shown in the top figure, the energy storage elements are two capacitors (labeled C1 and C2) and an inductor (labeled L; L1 in lower figure). A "locally active resistor" is a device that has negative resistance and is active (it can amplify), providing the power to generate the oscillating current. The locally active resistor and nonlinearity are combined in the device NR, which is called "Chua's diode". This device is not sold commercially but is implemented in various ways by active circuits. The circuit diagram shows one common implementation. The nonlinear resistor is implemented by two linear resistors and two diodes. At the far right is a negative impedance converter made from three linear resistors and an operational amplifier, which implements the locally active resistance (negative resistance). Analyzing the circuit using Kirchhoff's circuit laws, the dynamics of Chua's circuit can be accurately modeled by means of a system of three nonlinear ordinary differential equations in the variables x(t), y(t), and z(t), which represent the voltages across the capacitors C1 and C2 and the electric current in the inductor L1 respectively.