An elementary approach to dynamics and bifurcations of skew tent maps

Abstract

In this paper, the dynamics of skew tent maps are classified in terms of two bifurcation parameters. In time series analysis such maps are usually referred to as continuous threshold autoregressive models (TAR(1)-models) after Tong (Non-Linear Time Series, Clarendon Press, Oxford, UK, 1990). This study contains results simplifying the use of TAR(1)-models considerably, e.g. if a periodic attractor exists it is unique. On the other hand, we also claim that care must be exercised when TAR models are used. In fact, they possess a very special type of dynamical pattern with respect to the bifurcation parameters and their transition to chaos is far from standard.

Keywords:

• threshold autoregressive model
• skew tent map
• uniqueness of periodic attractors
• difference equation
• chaos
• bifurcation diagram

Keywords:

• 37E05
• 37E15
• 37G15
• 37G35
• 62M10
• 92D25

Acknowledgements

T. Lindström has been a holder of several grants during the completion of this paper. The first ideas were formulated as a part of a community training project financed by the European Commission through the Training and Mobility of Researchers (TMR) Programme. Further support from the Swedish Research Council and the Royal Swedish Academy of Sciences made it possible to complete this study. The authors thank professor Y. Rogovchenko for suggesting this journal as a forum for this discussion and an anonymous referee for comments regarding the manuscript.