Detecting chaos in a duopoly model via symbolic dynamics

Jose S. Cánovas; Tönu Puu; Manuel Ruiz Marín

doi:10.3934/dcdsb.2010.13.269

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2010, Volume 13, Issue 2: 269-278. Doi: 10.3934/dcdsb.2010.13.269

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Detecting chaos in a duopoly model via symbolic dynamics

1.
Dpto. Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Dr. Fleming s/n 30202, Cartagena
2.
Centre for Regional Science, Umeå University, SE-90187 Umeå
3.
Dpto. Métodos Cuantitativos e Informáticos, Universidad Politécnica de Cartagena, Paseo Alfonso XIII 50, 30203, Cartagena

Received: February 29, 2008

Revised: February 28, 2009

Published: August 2010

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Abstract

This paper considers a Cournot duopoly model assuming isoelastic demand and smooth cost functions with built-in capacity limits. When the firms cannot obtain positive profits they are assumed to choose small "stand-by" outputs rather than closing down, in order to avoid substantial fitting up costs when market conditions turn out more favorable. It is shown that the model provides chaotic behavior. In particular, the system has positive topological entropy and hence the map is chaotic in the Li-Yorke sense. Moreover, chaos is not only topological but also physically observable.

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Mathematics Subject Classification: Primary: 37N40, 37B55; Secondary: 37B40.

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