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**Chaotic sets and Euler equation branching.**
*(English)*
Zbl 1232.91467

Summary: Some macroeconomic models may exhibit a type of indeterminacy known as Euler equation branching (e.g., the one-sector growth model with a production externality). The dynamics in such models are governed by a differential inclusion \(\dot x \in H(x)\), where \(H\) is a set-valued function. In this paper, we introduce the concept of a chaotic set and explore its implications for Devaney chaos, Li-Yorke chaos and distributional chaos (adapted to dynamical systems generated by a differential inclusion). We show that a chaotic set will imply Devaney and Li-Yorke chaos and that a chaotic set with Euler equation branching will imply distributional chaos. We show that the existence of a steady state for a differential inclusion on the plane will generate a chaotic set and hence Devaney and Li-Yorke chaos. As an application, we show how these results can be applied to a one-sector growth model with a production externality – extending the results of Christiano and Harrison (1999). We show that chaotic (Devaney, Li-Yorke and distributional) and cyclic equilibria are possible and that this behavior is not dependent on the steady state being “locally” a saddle, sink or source.

### MSC:

91B52 | Special types of economic equilibria |

91B64 | Macroeconomic theory (monetary models, models of taxation) |

### Keywords:

indeterminacy; Euler equation branching; multiple equilibria; cycles; Devaney chaos; Li-Yorke chaos; distributional chaos; increasing returns to scale; externality; regime switching
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\textit{B. E. Raines} and \textit{D. R. Stockman}, J. Math. Econ. 46, No. 6, 1173--1193 (2010; Zbl 1232.91467)

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