Bifurcation routes to chaos in an extended van der Pol's equation applied to economic models

Abstract

In this paper a 3-dimensional system of autonomous differential equations is studied. It can be interpreted as an idealized macroeconomic model with foreign capital investment introduced in [9] or an idealized model of the firm profit introduced in [3]. The system has three endogenous variables with only one non-linear term and can be also interpreted as an extended van der Pol's equation. It's shown that this simple system covers several types of bifurcations: both supercritical and subcritical Hopf bifurcation and generalized Hopf bifurcation as well, the limit cycle exhibits period-doubling bifurcation as a route to chaos. Some results are analytical and those connected with chaotic motion are computed numerically with continuation programs Content, Xppaut and Maple. We present conditions for stability of the cycles, hysteresis, explore period doubling and using Poincare mapping show a three period cycle that implies chaos.

Keywords

• Hopf bifurcation
• period doubling
• chaos