Lozi map embedded into the 2D border collision normal form

Abstract

A 2D piecewise linear continuous two-parameter map known as the Lozi map is a special case of the 2D border collision normal form depending on four parameters. In the present paper, we investigate how the bifurcation structure of the Lozi map is incorporated into the bifurcation structure of the 2D border collision normal form using an analytical representation of the boundaries of the largest periodicity regions related to the cycles with rotation number 1/n, $n⩾3$. At the centre bifurcation boundary of the stability domain of the fixed point both maps are conservative which leads to a quite intricate bifurcation structure near this boundary.

Keywords:

• Lozi map
• 2D border collision normal form
• border collision bifurcation
• centre bifurcation
• degenerate bifurcations
• bifurcation structure

AMS Subject Classifications:

• 37E05
• 37G35
• 37N30

Acknowledgments

I. Sushko is grateful to the University of Urbino for its hospitality during her stay as a visiting professor as part of a project to support Ukrainian researchers during the Russian invasion of Ukraine. The work of V. Avrutin was supported by the German Research Foundation within the scope of the project ‘Generic bifurcation structures in piecewise-smooth maps with extremely high number of borders in theory and applications for power converter systems – 2’.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The meaning of the notion of normal form used here is different from that in standard bifurcation theory where it is related to locally topologically equivalent maps. The border collision normal form can be seen as a piecewise linear approximation of the piecewise smooth map in the neighbourhood of a border crossing fixed point.

2 In the considered case, the unique fixed point of the map F is a focus, and points of n-cycles we are dealing with are located around this fixed point. Simplifying, the rotation number of this cycle is an irreducible fraction m/n, m<n, where m is the number of rotations of the trajectory around the fixed point in n iterations. For a rigorous definition see [33].