Liénard equation
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.
Definition
Let and be two continuously differentiable functions on with an even function and an odd function. Then the second order ordinary differential equation of the formis called a Liénard equation.Liénard system
The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We definethen
is called a Liénard system.
Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution leads the Liénard equation to become a first order differential equation:
which is an Abel equation of the second kind.
Example
The Van der Pol oscillatoris 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 at small and positive otherwise. The Van der Pol equation has in general no exact, analytic solution. Such a solution for a limit cycle does exist if is a constant piece-wise function.