In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation[1] is a type of second-order ordinary differential equation named after the French physicist Alfred-Marie Liénard.
During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the uniqueness and existence of a limit cycle for such a system. A Liénard system with piecewise-linear functions can also contain homoclinic orbits.[2]
Let and be two continuously differentiable functions on with an even function and an odd function. Then the second order ordinary differential equation of the formis called a Liénard equation.
The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We define
F(x):=
| x | |
| \int | |
| 0 |
f(\xi)d\xi
x1:=x
x2:={dx\overdt}+F(x)
\begin{bmatrix}
| x |
1\\
| x |
2\end{bmatrix} =h(x1,x2):=\begin{bmatrix}x2-F(x1)\\ -g(x1) \end{bmatrix}
is called a Liénard system.
Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution
v={dx\overdt}
v{dv\overdx}+f(x)v+g(x)=0
which is an Abel equation of the second kind.[3] [4]
{d2x\overdt2}-\mu(1-x2){dx\overdt}+x=0
is a Liénard equation. The solution of a Van der Pol oscillator has a limit cycle. Such cycle has a solution of a Liénard equation with negative
f(x)
|x|
f(x)
f(x)
A Liénard system has a unique and stable limit cycle surrounding the origin if it satisfies the following additional properties:[6]
\limxF(x):=\limx
| x | |
| \int | |
| 0 |
f(\xi)d\xi =infty;