The structure of mode-locking regions of piecewise-linear continuous maps: II. Skew sawtooth maps

TL;DR

This paper demonstrates that near shrinking points in piecewise-linear maps, the dynamics can be effectively approximated by a three-parameter family of skew sawtooth circle maps, clarifying complex bifurcation patterns.

Contribution

It introduces a novel approximation of high-dimensional map dynamics using skew sawtooth maps within specific parameter sectors, enhancing understanding of bifurcation structures.

Findings

Skew sawtooth maps accurately approximate the dynamics near shrinking points.

The approximation error diminishes with proximity to the shrinking point.

The approach explains complex bifurcation patterns observed in the maps.

Abstract

In two-parameter bifurcation diagrams of piecewise-linear continuous maps on $\mathbb{R}^N$, mode-locking regions typically have points of zero width known as shrinking points. Near any shrinking point, but outside the associated mode-locking region, a significant proportion of parameter space can be usefully partitioned into a two-dimensional array of nearly-hyperbolic annular sectors. The purpose of this paper is to show that in these sectors the dynamics is well-approximated by a three-parameter family of skew sawtooth circle maps, where the relationship between the skew sawtooth maps and the $N$-dimensional map is fixed within each sector. The skew sawtooth maps are continuous, degree-one, and piecewise-linear, with two different slopes. They approximate the stable dynamics of the $N$-dimensional map with an error that goes to zero with the distance from the shrinking point. The results explain the complicated radial pattern of periodic, quasi-periodic, and chaotic dynamics that occurs near shrinking points.