Local dynamics for spherical optimal control problems

• Main Author: Brito, Paulo
• Publication Date: 1996
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10400.5/30512
• topic: Spherical Optimal Control Problems; Local Dynamics; Fold and Hopf Bifurcations; Habit Formation; Endogenous Time Preference

In this paper we present results on the characterization of the local dynamics for infinite horizon discounted continuous time spherical optimal control problem. Though the dimension of the problem is almost overwhelming, the structure of the jacobian of the variational system allow us to make a complete characterization only by computing the sums of the principal minors of orders two, four and six. Using function of these minors, which are coefficients of a cubic reduced characteristic polynomial, we present a complete taxonomy for local dynamics. By using geometrical methods, we found that the maximum dimension of the stable manifold is three and that several kinds of bifurcations can occur: fold, Hopf, double-fold and fold-Hopf. A model for a representative consumer with habit formation and endogenous rate of time preference by Shi and Epstein (1993) was used both as an application and as a mean to compare with alternative methods of characterization of local dynamics. Differently from those authors we were able to prove the possibility of existence of an Hopf bifurcation for low levels of the coefficient of risk aversion and of the rate of habit formation.