On a model of financial crisis: critical curves as new tools of global analysis

Abstract

Many models applicable to several fields are mathematically described by a nonlinear system of two difference equations, or map of the plane ℝ² into itself. This is particularly true in the economic context, where the variables often change at discrete times by their own nature or definition. Generally, in these models, the nonlinearities are such that the resulting map is one with a non-unique inverse, that is, an endomorphism. Examples Can be found in [1–4]. The model described in [3] is a particular case of the more general one presented in [4]. They interpret “economic cycles” and “financial crisis” endogenously generated from the nonlinear interaction between the “goods market” and the “money market”. We use this model to illustrate how new analytical tools, the critical curves, can be used to study (in endomorphisms) the local-global attractivity of fixed points, invariant curves, cycles or of other attracting sets (regular and chaotic), as well as to determine and characterize global bifurcations which cause changes in the structure of invariant sets, or in the structure of trajectories.