2D discontinuous piecewise linear map: Emergence of fashion cycles

L. Gardini, I. Sushko, K. Matsuyama

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


We consider a discrete-time version of the continuous-time fashion cycle model introduced in Matsuyama, 1992. Its dynamics are defined by a 2D discontinuous piecewise linear map depending on three parameters. In the parameter space of the map periodicity, regions associated with attracting cycles of different periods are organized in the period adding and period incrementing bifurcation structures. The boundaries of all the periodicity regions related to border collision bifurcations are obtained analytically in explicit form. We show the existence of several partially overlapping period incrementing structures, that is, a novelty for the considered class of maps. Moreover, we show that if the time-delay in the discrete time formulation of the model shrinks to zero, the number of period incrementing structures tends to infinity and the dynamics of the discrete time fashion cycle model converges to those of continuous-time fashion cycle model.

Original languageEnglish (US)
Article number055917
Issue number5
StatePublished - May 1 2018

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • General Physics and Astronomy
  • Mathematical Physics


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