Liénard equation

In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation [A. Liénard (1928) "Etude des oscillations entretenues," "Revue générale de l'électricité", vol. 23, pages 901-912 and 946-954.] 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

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

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:is called Liénard system.

Examples

* The Van der Pol oscillator $\{d^2x\; over\; dt^2\}-mu(1-x^2)\{dx\; over\; dt\}\; +x=\; 0$ 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;
* $lim\_\{x\; o\; infty\}\; F(x)\; :=\; lim\_\{x\; o\; infty\}\; int\_0^x\; f(t)\; dt\; =\; infty;$
* "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".

Footnotes

External links

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