Sunday, May 20, 2012

And I know that soon the sky will split And the planets will shift

Figure 1: A Business Cycle
I have been reporting some results from a bifurcation analysis of a formalization of Kaldor's 1940 model of the business cycle. Figures 1 and 2 illustrate the appearance of a business cycle with saddle-point stability in the Kaldor model. Suppose orbits like this arose in the dynamical system governing the solar system. Then the planets might form out of a nebular cloud. And the planets would whirl around their orbits for, maybe, millions of millenia. But then the planets will move away from their orbits, as the solar system falls apart.
Figure 2: National Output in The Business Cycle

By the way, my evidence for the existence of a limit cycle with saddle-point stability consists of graphical representations like these. I have not yet been able to find a sequence of (presumably 62) points that exactly repeat. Each of these points along such a limit cycle would have a corresponding stable and unstable set. And the possibility arises of these stable and unstable sets intertwining in a complicated fashion away from the limit cycle. The title of Agliari et al.'s paper refers to such homoclinic tangles of the stable and unstable sets of points along a limit cycle with saddle point stability. The title is not referring to a homoclinic bifurcation of a limit point at the origin, albeit they point out that bifurcation also.


  • Agliari, A.; R. Dieci; and L. Gardini (2007). "Homoclinic Tangles in a Kaldor-Like Business Cycle Model", Journal of Economic Behavior & Organization. V. 62: 324-347.

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