An Overview of Bifurcation Theory

In this appendix we want to provide a brief introduction and discussion of the concepts of dynamical systems and bifurcation theory which has been used in the preceding sections. We refer the reader interested in a more thorough discussion of the mathematical results of dynamical systems and bifurcation theory to the books of Wiggins (1990) and Kuznetsov (1995). A discussion intended to more economically motivated problems of dynamical systems and chaos can be found in Medio (1992) or Day (1994). It is noteworthy here that chaos is only possible in the nonautonomous case; that is, the dynamical system depends explicity on time; but not for the autonomous case where time does not enter directly in the dynamical system (cf. Wiggins (1990)). Since both models we have considered in the preceding sections are autonomous they cannot reveal any chaos. However, as the reader has seen, the dynamical systems we have derived reveal some bifurcations.