Liénard equation
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation[1] is a certain type of differential equation, named after the French physicist Alfred-Marie Liénard.
During the development of radio and vacuum tubes, Liénard equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the existence of a limit cycle for such a system.
Definition
Let f and g be two continuously differentiable functions on R, with g an odd function and f an even function then the second order ordinary differential equation of the form
is called Liénard equation. The equation can be transformed into an equivalent 2 dimensional system of ordinary differential equations. We define
then
is called Liénard system.
Examples
- The Van der Pol oscillator is a Liénard equation.
Liénard's theorem
A Liénard system has a unique and stable limit cycle surrounding the origin if it satisfies the following additional properties:
- g(x) > 0 for all x > 0;
- F(x) has exactly one positive root at some value p, where F(x) < 0 for 0 < x < p and F(x) > 0 and monotonic for x > p.
Applications
Recently,[2] the Liénard system has been shown to describe the operation of an optoelectronics circuit that uses a resonant tunnelling diode to drive a laser diode to make an optoelectronic voltage controlled oscillator.
Footnotes
- ^ A. Liénard (1928) "Etude des oscillations entretenues," Revue générale de l'électricité, vol. 23, pages 901-912 and 946-954.
- ^ Slight, T. J. et al., “A Liénard Oscillator Resonant Tunnelling Diode-Laser Diode Hybrid Integrated Circuit: Model and Experiment”,IEEE J. Quantum Electron., Vol. 44, No. 12, pp. 1158-1163, December 2008. http://dx.doi.org/10.1109/JQE.2008.2000924