In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation[1] 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.
Then the second order ordinary differential equation of the form
[3][4] The Van der Pol oscillator is a Liénard equation.
The solution of a Van der Pol oscillator has a limit cycle.
The Van der Pol equation has no exact, analytic solution.
[5] A Liénard system has a unique and stable limit cycle surrounding the origin if it satisfies the following additional properties:[6]