Publicação
Local dynamics for optimal control problems of 3-dimensional ODE systems
| Resumo: | This paper presents a complete characterization of the local dynamics for optimal control problems of 3-dimensional systems of ordinary differential equations, by using geometrical methods. We start by proving that the particular structure of the Jacobian implies that the 6th order characteristic polynomial is equivalent to a composition of two lower order polynomials, which are solvable by radicals. The classification problem for local dynamics is addressed by finding partitions, over an intermediate 3-dimensional space, which are homomorphic to the sub-spaces tangent to the complex, center and stable sub-manifolds. As main results, we get a local stability theorem and necessary conditions for the existence of fold, Hopf, double-fold and fold-Hopf bifurcations. Two particular applications are made: to a model with both habit formation and endogenous time preference, which was already studied by Shi and Epstein (1993), for comparing results; and to a two goods' habit formation model. We prove that, in the first case, a Hopf bifurcation may occur. |
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| Autores principais: | Brito, Paulo |
| Assunto: | Optimal Control Problems Local Dynamics fold and Hopf Bifurcations Habit Formation |
| Ano: | 1998 |
| País: | Portugal |
| Tipo de documento: | working paper |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade de Lisboa |
| Idioma: | inglês |
| Origem: | Repositório da Universidade de Lisboa |
| Resumo: | This paper presents a complete characterization of the local dynamics for optimal control problems of 3-dimensional systems of ordinary differential equations, by using geometrical methods. We start by proving that the particular structure of the Jacobian implies that the 6th order characteristic polynomial is equivalent to a composition of two lower order polynomials, which are solvable by radicals. The classification problem for local dynamics is addressed by finding partitions, over an intermediate 3-dimensional space, which are homomorphic to the sub-spaces tangent to the complex, center and stable sub-manifolds. As main results, we get a local stability theorem and necessary conditions for the existence of fold, Hopf, double-fold and fold-Hopf bifurcations. Two particular applications are made: to a model with both habit formation and endogenous time preference, which was already studied by Shi and Epstein (1993), for comparing results; and to a two goods' habit formation model. We prove that, in the first case, a Hopf bifurcation may occur. |
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