On non-smooth pitchfork bifurcations in invertible quasi-periodically forced 1-D maps

Abstract

In this note we revisit an example introduced by T. Jäger in which a Strange Non-chaotic Attractor seems to appear during a pitchfork bifurcation of invariant curves in a quasi-periodically forced 1-d map. In this example, it is remarkable that the map is invertible and, hence, the invariant curves are always reducible. In the first part of the paper we give a numerical description (based on a precise computation of invariant curves and Lyapunov exponents) of the phenomenon. The second part consists in a preliminary study of the phenomenon, in which we prove that an analytic self-symmetric invariant curve is persistent under perturbations.

Keywords:

• Symmetric invariant curves
• Newton method
• reducibility

AMS Subject Classifications:

• 37C55
• 37G40
• 37C60

Acknowledgements

The author F.J. Muñoz-Almaraz wants to thank André Vanderbauwhede for all the discussions about symmetry, which have always been highly fruitful.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

A. J. and J. C. T have been supported by the Spanish Ministerio de Economía y Competitividad / FEDER [grant number MTM2015-67724-P]; the Generalitat de Catalunya [grant number 2014 SGR 1145]. F. J. M.-A. has been supported by the Spanish Ministerio de Economía y Competitividad [grant number MTM2012-31821].