Abstract
In this paper, we show how a rich variety of dynamics may arise in a simple multiplier–accelerator model when a nonlinearity is introduced in the investment function. A specific sigmoidal functional form is used to model investments with respect to the variation in national income, in order to bound the level of investments. In fact, due to obvious material constraints, business strategies cannot sustain infinite investments. With the aid of analytical and numerical tools, we investigate the stability conditions, bifurcations, as well as periodic and chaotic dynamics for different specifications of the model that may include or not an endogenous government expenditure. We obtain some comparative statics results that shed light on the stabilizing or destabilizing role of the various parameters in the model. Globally, we study multistability phenomena, i.e., the coexistence of different kinds of attractors with a rich and complex dynamic structure.
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Notes
Or periodic oscillations with a very high period. In discrete-time maps, the Neimark–Sacker bifurcation occurs when a pair of eigenvalues cross the unitary circle of the complex plane. Thus, remarkable differences can be evidenced (with respect to the continuous time case), concerning the kind of motion along the closed invariant curve created at the bifurcation (it is no longer a unique trajectory but the closure of infinitely many distinct trajectories, either periodic or quasiperiodic) and the fate of such invariant curve as the parameters move far from their bifurcation values.
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Naimzada, A.K., Pecora, N. Dynamics of a multiplier–accelerator model with nonlinear investment function. Nonlinear Dyn 88, 1147–1161 (2017). https://doi.org/10.1007/s11071-016-3301-4
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DOI: https://doi.org/10.1007/s11071-016-3301-4