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A stylized model of business fluctuations is developed, where investment and debt accumulation are responsible for the endogenous dynamics of income. Despite the simplicity of the model, the resulting nonlinear, two-dimensional discrete-time dynamical system displays a wide range of possible dynamic outcomes. If a key parameter, representing the propensity to invest, shifts exogenously from low to high values, a transition across qualitatively different long-run scenarios is observed, associated to different levels of economic activity. Moreover, coexistence of attractors and path-dependence characterize the dynamics for an intermediate range of such a parameter. The impact of exogenous disturbances on such situations results in aperiodic time series subject to unpredictable booms and slumps.

In this paper a nonlinear discrete-time business cycle model of Kaldor-type is considered, to illustrate particular global bifurcations which determine the appearance or disappearance of attracting and repelling closed invariant curves. Such bifurcations sequences, which involve homoclinic tangencies and transversal intersection of the stable and unstable manifolds of saddle cycles, may increase the complexity of the basins of attraction of multiple, coexisting attractors. Particularly, for the business cycle model examined here, such dynamic phenomena explain the co-existence of two stable steady states and an attracting closed curve, with an intricate basin structure, for wide ranges of the parameters.

Empirical evidence has suggested that, facing different trading strategies and complicated decision, the proportions of agents relying on particular strategies may stay at constant level or vary over time. This paper presents a simple "dynamic market fraction" model of two groups of traders, fundamentalists and trend followers, under a market maker scenario. Market mood and evolutionary adaption are characterized by fixed and adaptive switching fraction among two groups, respectively. Using local stability and bifurcation analysis, as well as numerical simulation, the role played by the key parameters in the market behaviour is examined. Particular attention is paid to the impact of the market fraction, determined by the fixed proportions of confident fundamentalists and trend followers, and by the proportion of adaptively rational agents, who adopt different strategies over time depending on realized profits. © 2005 Elsevier Ltd. All rights reserved.