Liénard equation
From Wikipedia, the free encyclopedia
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a certain type of differential equation.
During the development of radio and vaccum 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.
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[edit] 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
- x1: = x
then
is called Liénard system.
[edit] Examples
- The Van der Pol oscillator
is a Liénard equation
[edit] Liénard's theorem
Given a Liénard system which satisfies the additional properties
- g(x) > 0 for all x > 0
- F(x) has exactly on positive root at p, F(x) < 0 for 0 < x < p and F(x) > 0 and monoton for x > p
then the Liénard system has a unique and Stability theory limit cycle surrounding the origin.
