Bifurcations and chaotic behavior in a simple model of the economic long wave

Abstract

Abstract This paper presents a formal stability analysis of a simplified Kondratieff wave model. For normal parameter values the model has a single unstable equilibrium point which, combined with nonlinear constraints in the model's table functions, creates a characteristic limit cycle behavior. For other parameter values, the model generates damped oscillations instead of the limit cycle or overwhelms the nonlinear constraints and exhibits sustained exponential growth or total collapse. By linear stability analysis we first determine the conditions for a Hopf bifurcation, the transition from a stable to an unstable equilibrium. 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 nonlinear functions fail to contain the system so that exponential runaway or collapse occurs. We then develop a DYNAMO program to calculate the Lyapunov exponents of the system during a simulation and discuss how these can be measures of the divergence or convergence of nearby trajectories. Finally, we illustrate how subsequent period doublings and chotic behavior can occur if the model is driven exogennously by a weak sine wave, representing, for instance, the shorterm business cycle.