In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order 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.
is called the Liénard equation.
The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We define
is called a Liénard system.
is a Liénard equation.
- g(x) > 0 for all x > 0;
- 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.