Limit cycles in generalized Liénard systems

This paper presents some new results which we obtained recently for the study of limit cycles of nonlinear dynamical systems. Particular attention is given to small limit cycles of generalized Liénard systems in the vicinity of the origin. New results for a number of cases of the Liénard systems are presented with the Hilbert number, H^(i,j)=H^(j,i), for j = 4, i = 10, 11, 12, 13; j = 5, i = 6, 7, 8, 9; and j = 6, i = 5, 6. Detailed proofs for the existence of limit cycles are given in four cases.