Existence of Limit Cycles and Control in Complete Keynesian System by Theory of Bifurcations - Torre, V. (Econometrica, 1977)
BETWEEN 1940 AND 1950, Kaldor [12], Goodwin [9] and Hicks [11] showed that adequate models of business cycle have to be essentially nonlinear, as only nonlinear systems allow The study of their properties is useful besides the construction of models, for example, the stochastic stability of the system, when shocks and external perturbations occur. This problem, already complex for linear systems, becomes even more complex for nonlinear systems (Kushner [15], Astrom [5]). Klein and Preston [13] and Kosobud and O'Neil [14] obtained interesting results on stochastic stability of nonlinear models of business cycles. Another interesting aspect is the dependence of the business cycle on the parameters characterizing the system. This analysis can be a sort of framework for the control of business cycles. The of stability can provide useful tools to this end. The of stability is a fusion of the two concepts of stability and qualitative behavior in the sense of topological equivalence. Andronov and Pontriagin [3] considered differential equations in two variables in a closed domain. They said that a system X is rough if, by perturbating it slightly (in the Cl-sense), one gets a system Y equivalent to X (in the sense specified in Appendix 1). Later Lefshetz [17] translated rough to structural stable. Thom [24, 25] saw stability, broadly understood, as the preservation of qualitative features under small perturbations. Smale [21, 22], Peixoto [19], and Abraham and Robbin [1] developed the giving fundamental theorems. Sotomayor [23], Andronov et al. [4], Chafee [6], and Sattinger [20] started to give good basis for the so called theory of bifurcations. Points of bifurcation are, in a parameter space, points where the topological structure changes abruptly, that is where stability fails: the creation of limit cycles from a multiple focus (Hopf bifurcation), the creation of a closed trajectory from a multiple limit cycle .... The of bifurcation can provide new criteria to prove the existence of limit cycles, besides the classical ones of Poincare and Bendixon. For example, it is possible to prove the existence of a limit cycle, without resorting to the theorem of Bendixon-Poincare, as done by Chang and Smyth [7] or to the theorem of Levison and Smyth, as done by Ichimura [16] for the Kaldor model.