Investigating robust chaos in piecewise-linear maps

Indranil Ghosh (UCD)

Time:

11AM Wednesday, 22 October 2025

Location:

UCD Confucius Institute, Room 1.01

The two-dimensional border-collision normal form is a family of piecewise-linear maps with complex bifurcation structures. In this talk, I will show how renormalisation as a technique can be utilised to better understand these structures. I will focus on the occurrence of robust chaos and how the topology (number of connected components) of the chaotic attractor changes in various regions where the map is orientation-preserving, orientation-reversing, or non-invertible. This is done via numerical evaluation of the renormalisation operator. Broadly speaking, renormalisation involves showing that, for some members of a family of maps, a higher iterate or induced map is conjugate to a different member of this family. I will show how the results obtained through renormalisation compare to those obtained with brute-force numerical computations using an algorithm of Eckstein and Avrutin. This method is based on computing the greatest common divisors in return times and agrees very well with our renormalisation approach. I will also talk about how to verify Devaney chaos in the orientation-preserving setting, and how robust chaos can be demonstrated in maps with more than two dimensions in terms of a positive Lyapunov exponent.