Li–Yorke chaos in higher dimensions: a review

Abstract

Li and Yorke not only introduced the term “chaos” along with a mathematically rigorous definition of what they meant by it, but also gave a condition for chaos in scalar difference equations, their equally famous “period three implies chaos” result. Generalizations of the Li and Yorke definition of chaos to difference equations in ℝ ⁿ are reviewed here as well as higher dimensional conditions ensuring its existence, specifically the “snap-back repeller” condition of Marotto and its counterpart for saddle points. In addition, further generalizations to mappings in Banach spaces and complete metric spaces are considered. These will be illustrated with various simple examples including an application to chaotic dynamics on the metric space (ℰ  ⁿ , D) of fuzzy sets on the base space ℝ ⁿ .

Keywords:

• Chaos
• Difference equations
• Metric space
• Saddle point
• Snap-back repeller

Acknowledgements

Peter Kloeden was partly supported by Ministerio de Educación y Ciencia (Spain) and FEDER (European Community) project MTM2005-01412 and Ministerio de Educación y Ciencia (Spain) grant SAB2004-0146 (Programa de Movilidad del Profesorado universitario español y extranjero). He thanks the Dpto. Ecuaciones Diferenciales y Análisis Numérico of the Universidad de Sevilla for its hospitality, during the time this article was prepared. Zhong Li was partly supported by the grant from the Germany Academic Exchange Service and the Research Grants Council of the Hong Kong Joint Research Scheme (Proj. Nr.: G_HK005/04).

Notes

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Additional information

Notes on contributors

Zhong Li

¶ ¶ Tel: [phone]. Fax: [phone].Email: [email protected]