On a Very Steep Version of the Standard Map

ABSTRACT

We consider the long-time behavior of the trajectories of the discontinuous analog of the standard Chirikov map. We prove that for some values of parameters all the trajectories remain bounded for all time. For other set of parameters we provide an estimate for the escape rate for the trajectories and present a numerically supported conjecture for the actual escape rate.

KEYWORDS:

• stability
• twist map
• switched systems

2000 AMS SUBJECT CLASSIFICATION:

• 37E05
• 37E10
• 37E40

Acknowledgments

Present work was done during the Summer@ICERM research program in 2015. Authors are deeply thankful to ICERM and Brown University for the hospitality and highly encouraging atmosphere. Authors also wish to thank Vadim Zharnitsky and Stefan Klajbor-Goderich for deep and fruitful discussions.