1. Introduction

This work aims to investigate and visualize as clearly as possible the dynamics of the so-called "dynastic cycle" model. The dynastic cycle models a three-class society of farmers, bandits and soldiers and is derived from Chinese history.

According to [Ush89] a dynastic cycle is a periodic alternation of the society between despotism and anarchy. As illustration he mentions the Chinese history: a new line of emperors creates a powerful regime of peace and property, but when the dynasty becomes older, a downswing takes place, which is accompanied by civil war, misery, poverty and population decline.

[FFP93] present a model very similar to the one introduced by [Ush89]. In my further work I want to refer to this second model. As in the first model, [FFP93] also investigate a model of society with three classes, which are the same as in the first system. The three class society consists of farmers who produce some goods, bandits who steal these goods, and of soldiers who try to protect the farmers by fighting the bandits, but also tax the farmers.

When talking about the "dynastic cycle", a periodic alternation between two states of the society is meant. The first state is arnarchy, when an equilibrium develops between bandits who steal and farmers who try to defend their goods, without any public authority. The second one is despotism, when soldiers tax the farmers and fight the bandits in order to protect their incomes from the tax revenue. Therefore soldiers punish bandits by execution.
In the first case (that is in the absence of any public authority) the model of [FFP93] supposes that farmers and bandits interact dynamically, according to a usual predator-prey relationship.

Most of the visualizations were done in the DynSys3D - System, a system that provides many different tools for visualizing analytically defined three-dimensional dynamical systems in the AVS System.

The rest of this work is organized as follows:

In Section 2 the model is explained by showing which parameters are corresponding to what features of the society model.
In Section 3 the model is simplified by rescaling and fixing some of the parameters, therefore decreasing the numbers of parameters, that have to be handled.
In Section 4 the interaction of farmers and bandits under the assumption of a fixed number of soldiers is investigated. A two-dimensional bifurcation analysis is provided and visualized.
In Section 5 the full model is explored, by assuming that the dynamics of the soldiers is slow compared whith that of the farmers and bandits. This assumption is not unrealistic, especially when investigating the history of several different societies.
Next, in Section 6 the slow-fast assumption is relaxed, and a numerical bifurcation analysis is carried out on the full three-dimensional system, showing, that there exist parameter settings, for which the system exhibits chaotic behaviour.
In Section 7 some Conclusions are drawn.
In the last few Sections some acknowledgments, references and interesting links are provided.

Last updated on May 2, 1997 by Helmut Doleisch (helmut@cg.tuwien.ac.at)