Speaker:
Oleg Makarenkov
Date of Talk:
11/18/10
Affiliation:
Imperial College, London
Title:
Linear and Piecewise Linear Maps in Studying Grazing Bifurcations of a Periodic Solution
Abstract
If a periodic solution of a differential equation touches (grazes) an impact surface (that breaks trajectories according to a certain law) then the response (grazing bifurcation) of this trajectory to perturbations is dramatic, including periodic solutions of arbitrary long periods and horseshoes. By introducing a convenient cross-section many authors arrived to a piecewise smooth discrete dynamical system that deemed to model the dynamics in the neighborhood of the grazing solution pretty well. This dynamical system is typically linear on one side of the switching surface and of a square-root type on the another side. I will address the fact, that for Hamiltonian differential equations this square-root term often disappears making the resulting discrete dynamical system just piecewise linear. This may make applicable recent results on the dynamics of piecewise linear maps to differential equations with impacts.
