A Homoclinic Bifurcation

Figure 1: Ascending and Descending

I have been presenting some results from an analysis of formalizations of Kaldor's 1940 model of the business cycle. This post illustrates some possible behaviors qualitatively similar to those already reported in the literature.

Figures 2, 3, and 4 display some orbits in the (normalized) state space of the Kaldor model, with variations in one parameter determining variations in the topology of these particular phase portraits. In each figure, a movement to the right along the x axis corresponds to an increase in the value of the economy's stock of capital. A movement upward along the y axis corresponds to an increase in the national income. The propensity to save is higher for each figure in the series, but the propensity to save is always small enough that three fixed points exist for the model. In all cases, the middle fixed point has the (in) stability of a saddle point.

Figure 2: Kaldor's Model without a Business Cycle

Figure 3: A Homoclinic Bifurcation in Kaldor's Model

Figure 4: A Business Cycle in Kaldor's Model

A saddle point is such that a ball starting in the direction of the horse's head or tail rolls downward to the center. The bright yellow orbit in each of the three figures represents such a trajectory. The yellow line is known as the stable set of the corresponding fixed point. A ball would have to be balanced just so to achieve such a trajectory on an actual saddle. A ball perturbed from the center of the saddle would tend to roll downward to either side of the horse. The light blue (cyan) orbit in Figures 2 and 4 represent such a trajectory, called the unstable set of the corresponding fixed point.

A bifurcation analysis identifies qualitative changes in the phase portraits for a dynamical system with variations in the system parameters. Several bifurcations exist between Figures 2 and 3, and, I think, two bifurcations arise between Figures 3 and 4. The stable and the unstable sets of the fixed point at the origin, in some sense, have switched roles in the illustrated bifurcations. In Figure 2, the unstable set shown flows from the origin to the other two fixed points. In Figure 4, the stable set flows backwards in time from the origin to the other two fixed points. Of course, some other global behavior is an important difference among these figures. For example, a business cycle does not exist in Figure 2, while Figures 3 and 4 both display a stable business cycle. In the language of dynamical systems, this business cycle is known as a (stable) limit cycle.

The stable and the unstable sets of the origin correspond in Figure 3. Such correspondence of these sets for a given fixed point (or, say, limit cycle) is known as a homoclinic bifurcation. Homoclinic bifurcations are global phenomena and cannot be identified by a merely local stability analysis of the given fixed point. Can you see why one might draw an analogy between a homoclinic bifurcation and the M. C. Escher etching I choose to head this post with?

References

• Agliari, A.; R. Dieci; and L. Gardini (2007). "Homoclinic Tangles in a Kaldor-Like Business Cycle Mode", Journal of Economic Behavior & Organization. V. 62: 324-347.
• Bischi, G. I.; R. Dieci; G. Rodano; and E. Saltari (2001). "Multiple Attractors and Global Bifurcations in a Kaldor-type Business Cycle Model", Journal of Evolutionary Economics. V. 11: 257-554.