Research projects

Project CEMAPRE internal

TitleLyapunov exponents of randomly perturbed billiards
ParticipantsGianluigi Del Magno, José Pedro Gaivão (Principal Investigator), João Lopes Dias
SummaryThis project deals with Lyapunov exponents, which is a quantity that measures the complexity of
dynamical systems. More precisely, Lyapunov exponents quantify the rate of separation of nearby
orbits, and positive Lyapunov exponents indicates that the system has sensitive dependence on
initial conditions, a phenomenon also known as the butterfly effect. Establishing positivity of
Lyapunov exponents of a given dynamical system is a highly non-trivial task. In most applications,
dynamical systems are subject to random external perturbations. If the unperturbed system has some
hyperbolicity, then random perturbations with absolutely continuous probability transitions are
expected to produce positive Lyapunov exponents. A notable class of dynamical systems are the
mathematical billiards. When the billiard table is strictly convex and non-integrable, it is long
standing open problem to prove the positivity of the top Lyapunov exponent.