CHAOS AND CHAOTIC TRANSIENTS IN A FORCED MODEL OF THE ECONOMIC LONG WAVE: THE ROLE OF HOMOCLINIC BIFURCATION TO INFINITY

Abstract

When an autonomous system of ordinary differential equations exhibits limit cycle behavior but is close in parameter space to a homoclinic bifurcation to infinity in which the limit cycle blows up to infinite amplitude and disappears, periodic forcing of the system may result in the appearance of both chaos and chaotic transients. In this paper, we use numerical techniques to map out Arnol’d tongues of a forced model of the economic long wave and illustrate how the system becomes chaotic and also exhibits chaotic transients for certain parameter combinations. Based on linearizations at infinity, we argue that infinity acts like a saddle with stable and unstable manifolds. By numerical computation, we show that chaotic transients occur when the manifolds intersect. Depending on parameters, two types of bifurcations have been identified: A chaotic attractor blows up to infinite size and disappears or the boundary of the basin of attraction of a periodic solution becomes fractal.