On the global dynamics of periodic triangular maps

Abstract

This paper is an extension of an earlier paper that dealt with global dynamics in autonomous triangular maps. In the current paper, we extend the results on global dynamics of autonomous triangular maps to periodic non-autonomous triangular maps. We show that, under certain conditions, the orbit of every point in a periodic non-autonomous triangular map converges to a fixed point (respectively, periodic orbit of period p) if and only if there is no periodic orbit of prime period two (respectively, periodic orbits of prime period greater than p).

Keywords:

• Periodic triangular maps
• periodic non-autonomous difference equations
• stability
• global dynamics
• applications

AMS Subject Classifications:

• 37C75
• 39A23
• 39A30

Notes

The authors declare that there is no conflict of interests regarding the publication of this paper.

1 Sharkovsky’s Theorem was established in 1964 by Alexander Sharkovsky in Russian [15], and later translated to English by J. Tolosa in 1995 [16]. It has played a role of paramount importance in the dynamics of one-dimensional maps.

Additional information

Funding

This work was partially supported by FCT/Portugal through project PEst-OE/EEI/LA0009/2013.