Introduction to Dynamical Systems - Lecture 2

Explore the fundamentals of non-smooth dynamical systems in this comprehensive lecture by Soumitro Banerjee from the International Centre for Theoretical Sciences. Delve into the study of bifurcations and their importance in understanding qualitative changes in dynamical systems. Learn about switched dynamical systems, their applications in various fields, and the concept of border collision bifurcation. Examine examples of hard-impacting systems, discrete-time mappings, and the characteristics of fixed points. Gain insights into the three basic bifurcations, including the Neimark-Sacker bifurcation, and understand how to obtain discrete-time mappings for practical systems. This lecture is part of the Dynamics of Complex Systems 2018 program, which focuses on non-smooth dynamical systems and complex networks.

Three basic bifurcations
An example
Neimark-Sacker bifurcation
Introduction to Non-smooth Dynamical Systems
Switched Dynamical Systems
State moves between compartments
State does not move between compartments, yet equations change
An Example: hard-impacting system
Character of a fixed point
Discrete-time mapping is piecewise smooth
The discrete-time map
Map and Jacobian elements continuous
Map discontinuous, and has square root singularity
Border collision bifurcation
Obtaining the discrete-time mapping: Example