Liénard equation

In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation 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

${d^{2}x \over dt^{2}}+f(x){dx \over dt}+g(x)=0$

is called Liénard equation. The equation can be transformed into an equivalent 2 dimensional system of ordinary differential equations. We define

$F(x):=\int _{0}^{x}f(t)dt$

$x_{1}:=x$

$x_{2}:={dx \over dt}+F(x)$

then

${\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}=h(x_{1},x_{2}):={\begin{bmatrix}x_{2}-F(x_{1})\\-g(x_{1})\end{bmatrix}}$

is called Liénard system.

Examples

• The Van der Pol oscillator ${d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0$ 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;
• $\lim _{x\to \infty }F(x):=\lim _{x\to \infty }\int _{0}^{x}f(t)dt\ =\infty ;$
• 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, 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.

External links

• LienardSystem at PlanetMath.