In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a type of second-order ordinary differential equation named after the French physicist Alfred-Marie Liénard.

During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the uniqueness and existence of a limit cycle for such a system. A Liénard system with piecewise-linear functions can also contain homoclinic orbits.

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

then

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:

which is an Abel equation of the second kind.