Lienard equation

The Lienard equation is an equation often used in the theory of oscillations and dynamical systems . Named after the French physicist A. Lienard .

Content

Definition

Let be $f$ ${\ displaystyle f}$ and $g$ ${\ displaystyle g}$ - two smooth functions in space $R^{3}$ ${\ displaystyle R ^ {3}}$ . Let be $g$ ${\ displaystyle g}$ Is an odd function , and $f$ ${\ displaystyle f}$ - even . Then an equation of the form

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

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

called the Lienard equation. ^[one]

In addition, the Lienard equation can be ^[2] ^[3] reduced to a first-order differential equation by making the change $v={dx \over dt}$ ${\ displaystyle v = {dx \ over dt}}$ . Then the Lienard equation is transformed into the Abel equation of the second type: $v{dv \over dx}+f(x)v+g(x)=0$ ${\ displaystyle v {dv \ over dx} + f (x) v + g (x) = 0}$

Examples

Oscillator Van der Pol ${d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0$ ${\ displaystyle {d ^ {2} x \ over dt ^ {2}} - \ mu (1-x ^ {2}) {dx \ over dt} + x = 0}$ has the form of the Lienard equation for $\left\{{\begin{matrix}f(x)=\mu (1-x^{2})\\g(x)=x\end{matrix}}\right.$ ${\ displaystyle \ left \ {{\ begin {matrix} f (x) = \ mu (1-x ^ {2}) \\ g (x) = x \ end {matrix}} \ right.}$ .

Related Definitions

Lienar system

The Lienard equation can be transformed into a system of differential equations .

Let be

F(x):=\int _{0}^{x}f(\xi )d\xi

{\ displaystyle F (x): = \ int _ {0} ^ {x} f (\ xi) d \ xi}

;

x_{1}:=x

{\ displaystyle x_ {1}: = x}

;

x_{2}:={dx \over dt}+F(x)

{\ displaystyle x_ {2}: = {dx \ over dt} + F (x)}

.

Then a system of the form

{\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}=\mathbf {h} (x_{1},x_{2}):={\begin{bmatrix}x_{2}-F(x_{1})\\-g(x_{1})\end{bmatrix}}

{\ displaystyle {\ begin {bmatrix} {\ dot {x}} _ {1} \\ {\ dot {x}} _ {2} \ end {bmatrix}} = \ mathbf {h} (x_ {1} , x_ {2}): = {\ begin {bmatrix} x_ {2} -F (x_ {1}) \\ - g (x_ {1}) \ end {bmatrix}}}

called the Lienar system.

Lienard's theorem

A Lienard system has a single and stable limit cycle near the origin if the system satisfies the following three criteria:

$g(x)>0$ ${\ displaystyle g (x)> 0}$ for all $x>0$ {\ displaystyle x> 0} ;
$\lim _{x\to \infty }F(x):=\lim _{x\to \infty }\int _{0}^{x}f(\xi )d\xi \ =\infty ;$ ${\ displaystyle \ lim _ {x \ to \ infty} F (x): = \ lim _ {x \ to \ infty} \ int _ {0} ^ {x} f (\ xi) d \ xi \ = \ infty;}$
$F(x)$ ${\ displaystyle F (x)}$ has only one positive root for some parameter value $p$ ${\ displaystyle p}$ where

F(x)<0

{\ displaystyle F (x) <0}

at

{\displaystyle 0<x<semantics><mrow class=

{\ displaystyle 0 <x <p}

and

F(x)>0

{\ displaystyle F (x)> 0}

and monotonous at

x>p

{\ displaystyle x> p}

.

Notes

↑ Liénard, A. (1928) "Etude des oscillations entretenues," Revue générale de l'électricité 23 , pp. 901–912 and 946–954.
↑ Liénard equation at eqworld .
↑ Abel equation of the second kind at eqworld .