6. What happens without the slow-fast hypothesis

In the model this can be achieved by relaxing the restriction, that R and G have to be very small. Under this new situation, the behavior of the model becomes much more complex and more and more different from that described in the previous section, so that a different approach to the analysis is needed. In this section, a numerical study of the bifurcations of the solutions of the system is presented. Especially the existence of chaotic behavior will be evidenced in a region of the parameter space.

In this part of the work the parameters E and R are considered as bifurcation parameters, while B, H, Q and D have been fixed as in chapter 4 and G = 0.09.

The (nonnegative) equilibria of the system are given by

The numerical analysis reveals that, at the above parameter settings, the solution of (6c) is unique. It collides with the equilibrium (6b) at E = B+1 (transcritical bifurcation).

In the next figure given below the bifurcation diagram of the system in a region of the parameter space (R,E) is presented. It has been derived by means of a continuation procedure ([Khi93], [FFP93]).

The nontrivial equilibrium given by (6c) is stable in the region "0" of the above bifurcation diagram, namely above the transcritical bifurcation curve TC (i.e., the line E = B+1). By increasing E, the equilibrium undergoes a Hopf bifurcation on the curve HF, so that in the region "1" a stable limit cycle exists, while the equilibrium has become unstable. This cycle then goes through a period-doubling bifurcation if the curve PD1 is crossed. Thus, in the region "2" the cycle that was generated by the Hopf bifurcation, is now unstable, while a stable cycle with the double period exists.

In the above bifurcation diagram also a second period-doubling curve PD2 is shown. By crossing PD2, the double period cycle generated by the first period-doubling bifurcation at PD1 loses stability, and a stable fourtimes-period cycle appears.
The PD2 curve is closed, and inside this curve other period-doubling closed curves (PD4, ...) have been detected next to PD2. This indicates the existence of a Feigenbaum's cascade [GuHo86]: thus, by increasing E while keeping R constant to a suitable value, the system experiences a sequence of period-doubling bifurcations leading to a value of E where the system enters the region of chaotic behavior. Then, by continuing to increase E, a reverse cascade takes place.

In the following pictures several visualizations are shown for different values of E with R fixed at R = 0.1.

In the next four pictures the way from the single stable cycle, that was created by the Hopf bifurcation, to the chaos over period-doubling is shown. The second picture shows the double-period cycle, the third the fourtimes-period cycle, that was created by crossing the PD2 line in the above bifurcation diagram.

In the following four pictures again four different stages are shown, but this time from the other direction. The first two pictures show the same scene as the first two of the four pictures above, but only from the opposite direction. The third picture shows the chaotic behavior from the above fourth picture, and the fourth pictures of the below ones shows the single period cycle, that one obtains, after having gone even further through the reverse cascades.

There is also an animation (~20MB) available, showing all the way from the single-period cycle over period-doubling to chaotic behavior and than via the reverse cascade back to a single-period cycle (be careful, that is a very large file!!!).

The next picture now shows an enhancement of the previous visualizations of the single-period cycle. Below the streamline visualizing the cycle, three timeplots are drawn, showing x, y, and z over the time. The Farmers (x) are shown by the red curve, the bandits (y) are given through the green curve, and the soldiers (z) finally are visualized by the blue curve in this timeplot. It has to be mentioned, that only a very short part over time is shown, but one can clearly recognize the cyclic behavior. And one can also clearly see, that most of the time, the system stays in a stable state (farmers top, bandits nearly 0, while the soldiers are reacting quite fast to any changes in these states).

In the next image the same single-period cycle is shown as in the picture above, but this time the three timeplots are not done in different colors (to be able to know which is the farmers, ...). Instead the original color of the colored streamline, that visualizes the cycle in this image was preserved, and therefore the velocity (which is encoded by the color) can be shown also in the timeplot. Here one can even better than before recognize, that most of the time (velocity is very low = blue) is spent in the stable state.

In the next three images the same as in the two above is shown, but this time for the double-period cycle. In the upper left image the timeplots are again distinguished by different colors, to be able to know which one is which. In the upper right image the velocity is again taken to colorcode also the timeplot. In the lower middle image finally, both types of timeplots are shown side by side, so the user has both advantages, he can distinguish the different system variables, but is also preserved the information, provided by the original colorcoding of the streamline showing the cycle.
In the timeplots of the below images the double-period of the cycle can be seen, there is two different patterns, which are repeated over time.

In the three pictures shown below the different timeplots introduced above are shown for the case of the fourtimes-period cycle.

In the next three images the same timeplots are shown for the chaotic behavior case. There is only one difference to the visualizations of the above cases: In the third picture (the lower one) there has been performed a zoom out, to show better, that the behavior is really chaotic, and not just in this short time.

In the next Section there will be some conclusions and an outlook to what else could be done to visualize the model of the dynastic cycle in the future.