Van der Pol oscillator - Scholarpedia. When x is small, the quadratic term x2 is negligible and the systembecomes a linear differential equation with a negative damping - epsilon dot x Thus, the fixed point x0, dot x 0, is un stable an unstable focus when 0 epsilon 2 and an unstable node, otherwise .On the other hand, when x is large, the term x2 becomes dominantand the damping becomes positive.Therefore, the dynamics of the system is expected to be restrictedin some area around the fixed point.Actually, the van der Pol system 1 satisfies the Liénard's' theorem ensuring that there is a stable limit cycle in the phase space The van der Pol system is therefore a Liénard system. |

Liénard equation - Wikipedia. Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution v d x d t displaystyle v dx over dt leads the Liénard equation to become a first order differential equation.: v d v d x f x v g x 0 displaystyle v dv over dx f x vg x 0 which belongs to Abel equation of the second kind. |

Periodic solutions of Liénard's' equation - ScienceDirect. ScienceDirect. Sign in Register. Download full issue. Journal of Mathematical Analysis and Applications. Volume 86, Issue 2, April 1982, Pages 379-386. Periodic solutions of Liénard's' equation. Author links open overlay panel Gabriele Villari. Add to Mendeley. https://doi.org/10.1016/0022-247X(82)90229-3: Get rights and content. |

LIÉNARD SYSTEMS. A great number of mathematical models of physical systems give riseto differential equations of the type. This is known as Liénard's' equation. The system is a generalizationof the mass-spring-damper system. In Liénard's' equation isthe damping term and isthe spring term. |