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Most real world phenomenon governed by deterministic laws of
physics can be modeled as a dynamical system. An abstract dynamical
system consists of a "phase-space" X and an homeomorphism $Phi^t:Xto
X$ called the flow, which changes continuously with time. Associated
to every dynamical system is a unitary map on its space of
observables, called the Koopman operator. This operator allows any
nonlinear system to be studied as a linear, unitary map in functional
space. The first part will be about the spectral properties of this
operator and its relevance to the underlying dynamics. Of particular
significance are the eigenfunctions of the Koopman operator, one among
many of their physical significance is that they correspond to stable
spatio-temporal patterns in the dynamics. The second part of the talk
will be about how these eigenfunctions can be extracted in various
ways from a time-series of some observation map $f$. This is called a
``data-driven" approach, which means inferring properties of the
system from a time-series which could be of dimension much smaller
than the underlying system, and without having any prior knowledge of the system, or a model or equations to start with, or parameters to
tune/fit. |