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Locating Local Bifurcations in Optimal Control Problems of 4-Dimensional ODE Systems

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  • Paulo Brito

    (ISEG - Technical University of Lisbon)

Abstract

The paper presents a complete characterization of the local dynamics for optimal control problems of 4-dimensional systems of ordinary differential equations, by using geometrical methods. We prove that the particular structure of the Jacobian implies that the 8 th 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 4-dimensional space, which are homomorphic to the sub-spaces tangent to the complex, center and stable sub-manifolds. Then we get local necessary conditions for the existence of 1- to 4-fold, Hopf, 1- and 2-fold-Hopf and Hopf-Hopf bifurcations, and represent them geometrically.

Suggested Citation

  • Paulo Brito, 2000. "Locating Local Bifurcations in Optimal Control Problems of 4-Dimensional ODE Systems," Econometric Society World Congress 2000 Contributed Papers 0506, Econometric Society.
    • Handle: RePEc:ecm:wc2000:0506
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    References listed on IDEAS

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    1. Feichtinger, Gustav & Novak, Andreas & Wirl, Franz, 1994. "Limit cycles in intertemporal adjustment models : Theory and applications," Journal of Economic Dynamics and Control, Elsevier, vol. 18(2), pages 353-380, March.
    2. Hartl, Richard F., 1987. "A simple proof of the monotonicity of the state trajectories in autonomous control problems," Journal of Economic Theory, Elsevier, vol. 41(1), pages 211-215, February.
    3. P.M.B. Brito, 1999. "Local dynamics for optimal control problemsof three‐dimensional ODE systems," Annals of Operations Research, Springer, vol. 89(0), pages 195-214, January.
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