Small-amplitude limit cycles of Liénard systems

Abstract

This paper is concerned with the study of second order differential equations of Lienard type: (A) $$\ddot x + f(x)\dot x + g(x) = 0$$ where f and g are polynomials. The equation (A) can also be written as a system of the form (B) $$\dot x = y - F(x),\dot y = - g(x),$$ , where $$F(x) = \mathop \smallint \limits_0^x f(\xi )d\xi $$ . The results described here are mainly concerned with small amplitude limit cycles; that is, limit cycles which may be bifurcated from the origin on perturbation of the coefficients of F and g. The problem is to estimate the maximum number of limit cycles which various classes of systems of the form (B) can have; this is a special case of the second part of Hilbert’s sixteenth problem. Most of the calculations have been carried out on a computer using the REDUCE symbolic manipulation package.