Chaotic Dynamics for Maps in One and Two Dimensions: A Geometrical Method and Applications to Economics

Abstract

This article describes a method — called here "the method of Stretching Along the Paths" (SAP) — to prove the existence of chaotic sets in discrete-time dynamical systems. The method of SAP, although mathematically rigorous, is based on some elementary geometrical considerations and is relatively easy to apply to models arising in applications. The paper provides a description of the basic mathematical ideas behind the method, as well as three applications to economic models. Incidentally, the paper also discusses some questions concerning the definition of chaos and some problems arising from economic models in which the dynamics are defined only implicitly.

Chaotic dynamics in the seasonally forced SIR epidemic model

“…Medio et al 2009) Consider (J , d) a metric space and D an open set. We say that a continuous map ψ : D −→ J induces chaotic dynamics on two symbols if there exist two disjoint compact sets K 0 , K 1 ⊂ D such that, for each two-sided sequence (s i ) i∈Z ∈ {0, 1} Z , there exists a corresponding sequence (ω i ) i∈Z ∈ (K 0 ∪ K 1 ) Z such that…”

“…Although numerical results have already suggested that the seasonal forced SIR model can exhibit chaotic dynamics, a rigorous proof was not given before. There are two main ingredients in our arguments: some subtle singular rescalings inspired by Smith (1983) and the method of stretching along the paths developed by Zanolin and his co-workers (Medio et al 2009;Margheri et al 2010Margheri et al , 2013Ruiz-Herrera and Zanolin 2014).…”

Chaotic social interaction via endogenous reactivity

“…one of the trademarks of chaos. Another attempt would consist in using the 'stretching along the paths' (from now on, SAP) method, already employed, for instance, in [12,17] to prove the existence of chaotic dynamics for both discrete and continuous-time dynamical systems. A third approach would consist in working with the first three iterates of map f in (2.7), in order to find suitable intervals where to apply Theorem 1 in [10] to prove the existence of chaos in the sense of Li -Yorke, as described in conditions (T1) and (T2) in that result.…”

“…Nonetheless, as seen in Proposition 4.1, the method in [7] still allows to draw some interesting conclusions from a dynamical viewpoint. For a more detailed comparison between the covering relations in [7,12], we refer the interested reader to [16].…”

Complex Dynamics in a ODE Model Related to Phase Transition

“…In Section 2 we recall the main topological tools which are used in the proof of our theorems. Namely, we give a brief survey of the so-called stretching along the paths (SAP) method introduced in [13] and further developed in a series of articles [12], [14], [15], [16]. Section 3 is devoted to the proof of Theorem 1 and to some of its immediate extensions.…”