Chua's circuit
Encyclopedia
Electronic circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow...
that exhibits classic chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
behavior. It was introduced in 1983 by Leon O. Chua
Leon O. Chua
Leon Ong Chua is an IEEE Fellow and a professor in the electrical engineering and computer sciences department at the University of California, Berkeley, which he joined in 1971...
, who was a visitor at Waseda University
Waseda University
, abbreviated as , is one of the most prestigious private universities in Japan and Asia. Its main campuses are located in the northern part of Shinjuku, Tokyo. Founded in 1882 as Tokyo Senmon Gakko, the institution was renamed "Waseda University" in 1902. It is known for its liberal climate...
in Japan
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
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."
Chaotic criteria
An autonomous circuitAutonomous Circuit
An autonomous circuit is a circuit that produces a time-varying output without having a time-varying input....
made from standard components (resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s, capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s, inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s) 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 figure, the energy storage elements are two capacitors (labeled C1 and C2) and an inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
(labeled L1). There is an active resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
(labeled R). There is a nonlinear resistor made of two linear resistors and two diodes. At the far right is a negative impedance converter
Negative impedance converter
The negative impedance converter is a one-port op-amp circuit acting as a negative load which injects energy into circuits in contrast to an ordinary load that consumes energy from them. This is achieved by adding or subtracting excessive varying voltage in series to the voltage drop across an...
made from three linear resistors and an operational amplifier
Operational amplifier
An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...
.
Model
By means of the application of the laws of electromagnetismElectromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, 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 give the voltages across the capacitors C1 and C2, and the intensity of the electrical current in the inductor L1, respectively. These equations read:
The function f(x) describes the electrical response of the nonlinear resistor, and its shape depends on the particular configuration of its components. The parameters α and β are determined by the particular values of the circuit components.
A chaotic attractor, known as "The Double Scroll" because of its shape in the (x,y,z) space, was first observed in a circuit containing a nonlinear element such that f(x) was a 3-segment piecewise-linear function.
The easy experimental implementation of the circuit, combined with the existence of a simple and accurate theoretical model, makes Chua's circuit a useful system to study many fundamental and applied issues of chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
. Because of this, it has been object of much study, and appears widely referenced in the literature.
Moreover the Chua' s circuit can be easily realized by using a multilayer CNN (Cellular Nonlinear Networks). The CNN have been just invented by Leon Chua in 1988. To date, а huge number of various types of chaotic attractors in Chua's system has been discovered, which can be obtained numerically, with relative ease, by standard computational procedure (after transient process a trajectory, started from a point of unstable manifold in a small neighborhood of unstable zero equilibrium, reaches an attractor and computes it). Also, recently, a hidden Chua's attractor
Hidden oscillation
From computation point of view, in nonlinear dynamical systems periodic oscillations and chaotic attractors can be regarded as self-exciting attractors or hidden attractors....
was discovered.
See also
- Interactive Chua's circuit 3D simulation Double Scroll Example
- MemristorMemristorMemristor is a passive two-terminal electrical component envisioned by Leon Chua as a fundamental non-linear circuit element relating charge and magnetic flux linkage...
Books
- Chaos synchronization in Chua's circuit, Leon O Chua, Berkeley : Electronics Research Laboratory, College of Engineering, University of California, [1992], OCLC: 44107698
- Chua’s Circuit Implementations: Yesterday, Today and Tomorrow,L. Fortuna, M. Frasca, M.G. Xibilia, World Scientific Series on Nonlinear Science, Series A - Vol. 65, 2009, ISBN 978-981-283-924-4
External links
- Chua's Circuit: Diagram and discussion
- NOEL laboratory. Leon O. Chua's laboratory at the University of California, Berkeley
- References
- Chua and Memristors
- Hidden attractor in Chua's system
- http://www.eecs.berkeley.edu/~chua/papers/Arena95.pdf