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Differential game model of the dynastic cycle: 3D-canonical system with a stable limit cycle


Ancient Chinese history reveals many examples of a cyclical pattern of social development connected with the rise and the decline of dynasties. In this paper, a possible explanation of the periodic alternation between despotism and anarchy by a dynamic game between the rulers and the bandits is offered. The third part of the society, the farmers, are dealt with as a renewable resource which is exploited by both players in a different manner. It is shown that the Nash solution of this one-state differential game may be a persistent cycle. Although we restrict the analysis to open-loop solutions, this result is of interest for at least two reasons. First, it provides one of the few existing dynamic economic games with periodic solutions. Second, and more important, the model is an example of a three-dimensional canonical system (one state, two costates) with a stable limit cycle as solution. As far as we see, our model provides up to now the simplest (i.e., lowest dimensional) case of a persistent periodic solution of an intertemporal decision problem.

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This research was partially supported by the Austrian Science Foundation under Contract No. P7783-PHY.

Helpful comments of T. Basar, E. J. Dockner, R. F. Hartl, A. Mehlmann, G. Sorger, and F. Wirl as well as the generous help of George Leitmann are gratefully acknowledged.

Communicated by G. Leitmann

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Feichtinger, G., Novak, A.J. Differential game model of the dynastic cycle: 3D-canonical system with a stable limit cycle. J Optim Theory Appl 80, 407–423 (1994).

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Key Words

  • Differential games
  • open-loop Nash solutions
  • Hopf bifurcations
  • stable limit cycles