Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

Viktor Avrutin,Fabio Tramontana,Iryna Sushko

doi:10.1155/2014/401319

Abstract

We study the bifurcation structure of the parameter space of a 1D continuous piecewise linear bimodal map which describes dynamics of a business cycle model introduced by Day-Shafer. In particular, we obtain the analytical expression of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and related fin structure). By crossing these boundaries the map displays robust chaos.

Highlights

Applied models de...ned by piecewise smooth functions appear quite often when one studies a real process characterized by some ‘nonsmooth’ phenomena such as sharp switchings between several states, impacts, friction, sliding, and the like
In this work we have considered a pioneering model by Day and Shafer [9] which describes a business cycle by using a bimodal piecewise linear map
Our investigation shows how rich is the dynamic behaviors of the system, going from attracting cycles of any period, to robust chaotic intervals, depending on the parameters values

Summary

Introduction

Applied models de...ned by piecewise smooth functions appear quite often when one studies a real process characterized by some ‘nonsmooth’ phenomena such as sharp switchings between several states, impacts, friction, sliding, and the like. A one-dimensional (1D for short) continuous piecewise linear map with one border point, known as skew tent map, depending on the parameters values can have attracting cycles of any period as a well as cyclic chaotic intervals of any period, called n bands chaotic attractors, which have the relevant property of being robust (as introduced in [3]) with respect to parameter perturbations. Recall that a BCB occurs when an invariant set, such as, for example, a ...xed point or cycle, collides with a border separating regions of di¤erent de...nition of the map. This bifurcation may lead, for example, from an attracting cycle directly to chaos.

The Day-Shafer model

Preliminaries

Two-dimensional bifurcation diagrams

Fin structure

Conclusions