Liénard's theorem

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In mathematics, more specifically in dynamical systems, Liénard's theorem guarantees the existence of a limit cycle for a system if it can be described in the form

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

and the functions $f (x)$ and $g (x)$ meet the following conditions:

$f (x)$ and $g (x)$ are continuously differentiable for all x
$g (x)$ is an odd function
$f (x)$ is an even function
$g (x) > 0$ for all $x > 0$
There must be an odd function $F(x) = \int^x_0 \!f(u)\, du$ where $F (x) < 0$ for $0 < x < a$ , $F (a) = 0$ , $F (x)$ is never decreasing for $x > a$ , and $F(x) \rightarrow \infty$ as $x \rightarrow \infty.$