Liénard equation

Let f and g be two continuously differentiable functions on ${\displaystyle \mathbb {R} ,}$ with f an even function and g an odd function. Then the second order ordinary differential equation of the form

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

is called a Liénard equation.

The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We define

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

${\displaystyle x_{1}:=x}$

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

then

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

is called a Liénard system.

Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution ${\displaystyle v={dx \over dt}}$ leads the Liénard equation to become a first order differential equation:

${\displaystyle v{dv \over dx}+f(x)v+g(x)=0}$

which is an Abel equation of the second kind.[3][4]

The Van der Pol oscillator

${\displaystyle {d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0}$

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 ${\displaystyle f(x)}$ at small ${\displaystyle |x|}$ and positive ${\displaystyle f(x)}$ otherwise. The Van der Pol equation has no exact, analytic solution. Such solution for a limit cycle exists if ${\displaystyle f(x)}$ is a constant piece-wise function.[5]

A Liénard system has a unique and stable limit cycle surrounding the origin if it satisfies the following additional properties:[6]

• g(x) > 0 for all x > 0;
• ${\displaystyle \lim _{x\to \infty }F(x):=\lim _{x\to \infty }\int _{0}^{x}f(\xi )d\xi \ =\infty$ ;}
• 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.

• Autonomous differential equation
• Abel equation of the second kind
• Biryukov equation

• "Liénard equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• LienardSystem at PlanetMath.