Liénard equation

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In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation^[1] is a certain type of differential equation.

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.

1 Definition
2 Examples
3 Liénard's theorem
4 Footnotes
5 External links

[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

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

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.

[edit] Examples

The Van der Pol oscillator ${d^2x \over dt^2}-\mu(1-x^2){dx \over dt} +x= 0$ is a Liénard equation.

[edit] 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.