Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 14, Iss. 1, January, 2010, pp. 69-83
@2010 Society for Chaos Theory in Psychology & Life Sciences

 
 
 

Numerical Explorations of R. M. Goodwin”s Business Cycle Model

Aleksander Jakimowicz, Nicolaus Copernicus University, Poland

Abstract: Goodwin”s model, which was formulated in 1951, still attracts economists” attention. The model possesses numerous interesting properties that have been discovered only recently due to the development of the chaos theory and the complexity theory. The first numerical explorations of the model were conducted in the early 1950s by Strotz, McAnulty and Naines (1953). They discovered the coexistence of attractors that are well-known today, two properties of chaotic systems: the sensitive dependence on the initial conditions and the sensitive dependence on parameters. The occurrence of periodic and chaotic attractors is dependent on the value of parameters in a system. In case of certain parametric values fractal basin boundaries exist which results in enormous system sensitivity to external noise. If periodic attractors are placed in the neighborhood of the fractal basin boundaries, then even a low external noise can move the trajectory into the region in which the basin’s structure is tangled. This leads to a kind of movement that resembles a chaotic movement on a strange attractor. In Goodwin”s model, apart from typical chaotic behavior, there exists yet another kind of complex movements - transient chaotic behavior that is caused by the occurrence of invariant chaotic sets that are not attracting. Such sets are represented by chaotic saddles. Some of the latest observation methods of trajectories lying on invariant chaotic sets that are not attracting are straddle methods. This article provides examples of the basin boundary straddle trajectory and the saddle straddle trajectory. These cases were studied by Lorenz and Nusse (2002). I supplement the results they acquired with calculations of capacity dimension and correlation dimension.

Keywords: nonlinear business cycle model, basins of attraction, fractal basin boundaries, chaotic saddle, BST method, bisection method, SST method, PIMtriple refinement procedure, basin boundary straddle trajectory, saddle straddle trajectory