On Some Periodic Solutions of the Lienard Equation

Oscillations described by the generalized Li6nard equation\ddot{x} + f(x)\dot{x} + g(x) = 0(\cdot = d/dt)are investigated in the Liénard plane. Whenf(x),g(x)andF(x)= \int_{0}^{x}f(\zeta)d\zetaare subject to certain restrictions, a number of analytic curves can be obtained in this plane which serve as bounds on solution trajectories. Piecewise connection of such bounding curves provides explicit annular regions with the property that solution trajectories on the boundary of an annulus move to the interior with increasing time,t. The Poincaré-Bendixson theorem then guarantees that at least one periodic orbit exists within such an annulus. Particular attention is given to damping functions,f(x), which are asymmetric.