By taking account of obvious and inescapable limitations on the functioning of the accelerator, we explain some of the chief characteristics of the cycle, notably its failure to die away, along with the fact that capital stock is usually either in excess or in short supply. By a succession of increasingly complex models, the nature and methods of analyzing nonlinear cycle models is developed. The roles of lags and of secular evolution are illustrated. In each case the system's equilibrium position is unstable, but there exists a stable limit cycle toward which all motions tend.