Resolution of chaos with application to a modified Samuelson model

Recently, several discrete nonlinear growth models with complicated dynamical behavior have been introduced in the literature. A great deal of these papers are numerically oriented and claim to establish chaotic dynamics (showing the existence of a periodic three-cycle) without elaborating very much on the mathematical aspects of the involved maps. We will see that for a suitable class of discrete processes there is order in the erratic dynamics. In fact, the dynamics of these simple processes will be dominated by ultimately periodic behavior, and we will show that this behavior is persistent under small smooth perturbations.

To illustrate bifurcation phenomena, we will consider a modified Samuelson model [a nonlinear multiplier-accelerator model introduced by Gabisch (1984)]. We will show that period-doubling bifurcation as well as period-halving bifurcation can occur as a parameter value is increased. For this model Gabisch showed that for a certain range of parameter values there is chaos in the dynamics in the sense of Li and Yorke. We will not only present results concerning the regularity in the complicated dynamics, but also we will show that the range of parameter values k, for which there is chaos in the sense of Li and Yorke, is small. Finally, the chaos will disappear when the accelerator is increased.

Furthermore, we will give an example (a modified Cobweb model) in which such period-doubling bifurcations do not occur. In particular, for these kinds of models, as soon as the equilibrium is unstable, there exist infinitely many periodic points with different period and there exist also aperiodic points.