Sydney Dynamics Group Seminar: Gary Froyland, University of New South Wales -- Sydney Dynamics group Seminar

Seminar : Sydney Dynamics group Seminar 
Day/time : Thursday, May 21 at 4:00 PM, 
Location: SMRI seminar room (A12 Macleay Room 301) 
Speaker : Gary Froyland, University of New South Wales 

Title: On the structure of complex spectra and eigenfunctions of transfer and Koopman 
operators 

(joint work with Matheus M.  Castro).  

Abstract: Complex eigenspectra of transfer and Koopman operators describe rotational
motion in dynamical systems.  A particularly relevant situation in applications is when
the rotation speed depends on the state-space position of the dynamics.  We consider a
canonical model of such dynamics in the presence of small noise, and provide precise
characterisations of the eigenspectrum and eigenfunctions of the corresponding transfer
operators.  Further, we study the limiting behaviour of the eigenspectrum and
eigenfunctions in the zero-noise limit, including their quadratic and linear response.
Our results clarify the structure of transfer and Koopman operator eigenspectra, and
provide new interpretations relevant to applications.  Our theorems on support
localisation of the eigenfunctions yield simple algorithms to detect the existence and
state-space location of approximately cyclic motion with distinct periods.  Our
numerical results verify that information on the cycle periods and their locations
determined by the operator eigendata is insensitive to noise level in the linear
response regime.  We believe that the dynamic mechanisms underlying the eigendata and
their properties apply rather broadly and enhance our understanding of approximate cycle
detection in dynamical systems with operator methods.