This paper presents a total stability analysis of a simplified Kondratieff-wave model. The purpose is to show how such an analysis can be carried out and to illustrate the kind of information one obtains. For normal parameter values the Kondratieff wave model has a single unstable equilibrium point. Combined with non-linear constraints in the model's table-functions, this instability creates a characteristic limit cycle behavior. For other parameter values, however, the model is stable and generates damped oscillations instead of the limit cycle. For yet other combinations of parameters, the non-linear constraints yield to the instability, and sustained exponential growth or total collapse result. By means of linear stability analysis we first determine the conditions for the transition between a stable and an unstable equilibrium to take place. This transition is known as a Hopf-bifurcation. Using global analysis we outline the phase-portrait of a fully developed limit cycle. By the same method, we examine the conditions under which the non-linear functions fail to contain the system so that exponential run-away or collapse occur. A DYNAMO-program is then developed which calculates the Lyapunov exponents of the system during a simulation, and we discuss how these exponents can be used as a measure of the divergence or convergence of nearby trajectories. Finally, we illustrate how subsequent period doublings and chaotic behaviour can occur if the model is driven exogenously by a weak sine-wave, representing for instance the short term business cycle.

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• 1984

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• Document

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System Dynamic Society Records

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