|Figure 1: Definition of A Poincaré Return Map|
Poincaré return maps are a useful tool for analyzing the local stability of limit cycles of a dynamical system. In this post, I explain how I define such a map for the Kaldor model of business cycles. I use the Poincaré return map to explain both the existence of a certain limit cycle with saddle-point stability and aspects of a sequence of bifurcations in the model. (My analysis of the Kaldor model also includes these two posts.) I have questions both about the rigorous definition of the Poincaré return map for a discrete-time system and about the consistency of my results with those in Agliari et al. (2007).
2.0 The Domain
The domain of the Poincaré return map, as I define it for the Kaldor model, is a line segment in the phase space. The upper left part of Figure 1 illustrates the phase space. The abscissa is the normalized value of the capital stock, and the ordinate is national income. Notice the line segment sloping upward to the right. It starts at the fixed point at the origin, and goes through the fixed point in the first quadrant. The domain for the map, as illustrated, is the line segment starting at this fixed point and continuing the ray from the origin an arbitrary distance upward. The argument of the Poincaré return map is then the distance along this line segment, with a value of zero corresponding to the fixed point.
Aside: The domain of the Poincaré return map can be used to define a split function1, useful in locating a homoclinic bifurcation of the origin, in the Kaldor model. (See Figure 3 here.) Let
- xs be the point in the domain of the Poincaré return function for the first crossing of the stable set of the origin with this domain, when the orbits comprising the stable set are followed backward in time.
- xu be the point in the domain of the Poincaré return function for the first crossing of the unstable set of the origin with this domain, when the orbits comprising the stable set are followed forward in time.
β = xu - xsβ is a function of the parameters of the Kaldor model. This is a split function, and its value is zero when a homoclinic bifurcation of the origin occurs. End of aside.
3.0 Definition of the Value of the Function
The Poincaré return map defines a discrete time dynamical system with one dimension less than the dynamical system, either continuous or discrete time, for which it is defined. The argument defines a point on the line segment in phase space corresponding to the domain of the map. Imagine an orbit in phase space starting at that point. Follow this orbit until, in this case, it crosses the extended line segment, defining the domain, from above. The point of intersection for this first crossing defines the value of the Poincaré return map for the given argument. It is the distance along this line segment from the fixed point in the first quadrant.
Since the Kaldor model is a discrete time dynamical system, an orbit starting on the line segment defining the domain of the map will likely not have a point on this line segment for the first crossing of the domain. This is not a problem for the numerical computation of the map; the computer program can calculate the intersection of the domain with the two points on the orbit straddling the line segment for the domain. I think this calculation of such intersections is the cause of the small wave-like ripples in the plot of the Poincaré return map in Figure 1. This likely failure of the points on an orbit to lie on the line segment defining the domain for a given positive number of crossing does raise a question in my mind about how to rigorously define the Poincaré return map for a discrete time dynamical system. Perhaps the map is only well-defined for a discrete set of parameter and argument values. I also wonder if the map will miss interesting orbits in phase space2.
A line sloping upward at 45 degrees is plotted in red in Figure 1 along with the Poincaré return map. Intersections of the map with this line are fixed points for the Poincaré return map. A fixed point corresponds to a limit cycle in phase space. The slope of the map at these intersections reflects the stability of the corresponding limit cycle:
- If the Poincaré return map is tangent to the 45o line, the corresponding limit cycle in phase space has the stability of a saddle-point.
- If the Poincaré return map slopes upward steeper than the 45o line at the point of intersection, the corresponding limit cycle is unstable.
- If the 45o line slopes upward steeper than the Poincaré return map at the point of intersection, the corresponding limit cycle is stable.
I ignored my qualms about the definition of the Poincaré return map and proceeded with an analysis of limit cycles in the Kaldor model. Figure 2 shows the location of fixed points of the map for an increasing savings rate and certain specified values of the remaining model parameters. No limit cycles exist for small values of the savings rate. A business cycle with the stability of a saddle point appears at a certain value of the savings rate. This limit cycle bifurcates into a stable and an unstable business cycle for higher values of the savings rate. Thus, Figure 2 is a bifurcation diagram for a fold bifurcation of the dynamical system defined by the Poincaré return map.
|Figure 2: A Fold Bifurcation of The Poincaré Return Map|
It seems to me that the above explains3 how a stable and unstable limit cycle arise in the Kaldor model, for example in Figure 3. Agliari et al. (2007) give a different explanation involving Arnold tongues and homoclinic bifurcations of the points comprising certain low-period orbits in phase space with saddle point stability. They claim that a number of such low-period orbits, with changing stability, appear in a sequence of bifurcations in the Kaldor model4. I do not see such orbits in the Poincaré return map for the region of parameter space that I have analyzed here. Figure 2 illustrates a local stability analysis, and homoclinic bifurcations seem to be only apparent in a global analysis. I do not understand how my analysis relates to theirs, albeit Agliari et al. state that much of the complexity that they analyze disappears in a final bifurcation in the sequence they describe.
|Figure 3: A Stable and Unstable Limit Cycle in the Kaldor Model|
- Figure 6.4, p. 198, in Section 6.1 of Kuznetsov (1998) illustrates a split function.
- Figure 4.10, p. 129, in Section 4.6 of Kuznetsov (1998) illustrates points in a low-period stable limit cycle and a saddle cycle alternating in a orbit arising in a Neimark-Sacker bifurcation. It is not clear to me how both orbits would show up in my numerically-calculated plots of the Poincaré return map.
- Figure 5-13, in Section 5.3 of Kuznetsov (1998) illustrates a fold bifurcation of a Poincaré return map and its connection with limit cycles in phase space.
- These bifurcations, I guess, are related to a Neimark-Sacker bifurcation of the fixed point at the origin.
- 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.
- Kuznetsov, Y. A. (1998). Elements of Applied Bifurcation Theory, Second edition. Springer-Verlag.