IMCS/Publications/BASM/Issues/BASM n.3 (40), 2002/

Static bifurcation diagram for a microeconomic model. (English)

Authors: Ungureanu Laura, Georgescu Adelina, Popescu Mariana.


A mathematical model governing the dynamics of the capital and working force of a firm is investigated. It consists in a Cauchy problem for a system of two nonliniar ordinary differential equations (O.D.E) depending on three real parameters. The closed form of the equilibrium points and their associated eigenvalues are deduced. A special attention is paid to nonhyperbolic equilibria and to situations when they coexist with other equilibria (hyperbolic or not). The degenerated cases corresponding to one, two or three null parameters are completely analyzed even from the dynamical viewpoint. In these situations the static and (partially) dynamic bifurcation diagrams are studied in the linear and nonlinear cases. It is found that nondegenerated or degenerated Hopf and Bogdanov-Takens bifurcations could occur. All these results allowed us to determine the static bifurcation diagram for the given model and to prepare the dynamic bifurcation diagram (consisting of the parameter portrait and the corresponding topologically nonequivalent phase portraits), which will be presented elsewhere. Among the results preliminary to the dynamical study in the parameters space we delimited the zones of the structural stability and of codimension -one, -two and -three bifurcations. All these zones followed from a local study about the nonhyperbolic equilibria. Further semilocal and global studies are expected to increase the codimension and to introduce additional strata. They will be done elsewhere.

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