Estimate for the amplitude of the limit cycle of the Liénard equation

Abstract

We consider the nonlinear Liénard equation \(\ddot x\left( t \right) + f\left( x \right)\dot x\left( t \right) + g\left( x \right) = 0\). Liénard obtained sufficient conditions on the functions f(x) and g(x) under which this equation has a unique stable limit cycle. Under additional conditions, we prove a theorem that permits one to estimate the amplitude (the maximum value of x) of this limit cycle from above. The theorem is used to estimate the amplitude of the limit cycle of the van der Pol equation \(\ddot x\left( t \right) + \mu \left[ {{x^2}\left( t \right) - 1} \right]\dot x\left( t \right) + x\left( t \right) = 0\).