This week AIMdyn’s team of scientists will present their research at the 2017 SIAM Conference on Applications of Dynamical Systems.
The conference will be held from May 21 to May 25 and features presentations from a large number of scientists who are experts in their respective fields.
Featured below we have a brief description of our teams’ material for the conference as well as their speaking times.
Extensions of Koopman operator theory: Stochastic Dynamical Systems and Partial Differential Equations
Speaker: Igor Mezic
Talk time: Tuesday, May 23, 8:30 – 8:55 AM
Symposium: MS79 Koopman Operator Techniques in Dynamical Systems: Theory
Abstract: We extend the theory of Koopman operators to the case of random dynamical systems. We show that the stochastic Koopman operator for random linear systems, defined using expectation over the state space of the underlying random process, admits eigenfunctions related to expectation of eigenvectors of the system. We use this to, via conjugacy, provide a theory for a large class of random dynamical systems. We also describe an extension of the Koopman operator theory for partial differential equations, where we define the notion of eigenfunctionals of the Koopman operator and describe consequences for reduced order modeling of the dynamics.
Koopman Spectrum for Cascaded Dynamical Systems
Speaker: Ryan Mohr
Co-author: Igor Mezic
Talk time: Tuesday, May 23, 9:00 – 9:25 AM
Symposium: MS79 Koopman Operator Techniques in Dynamical Systems: Theory
Abstract: We investigate the Koopman operator spectrum for cascaded dynamical systems – systems constructed by wiring subsystems together such that downstream systems do not influence upstream systems. Under precise conditions on the spectrums of the subsystems and norms of the matrices, we show that the cascaded system is asymptotically equivalent – with zero asymptotic relative error – to the decoupled system started from a perturbed initial condition. We use these results to show that the Koopman principal eigenfunctions of each subsystem can be extended to eigenfunctions for the cascaded system by composing them with the perturbation function. Using a topological conjugacy argument, we show that these results hold for nonlinear cascaded systems as well.
Baroreflex Physiology Using Koopman Mode Analysis
Speaker: Maria Fonoberova
Co-authors: Igor Mezic, Senka Macesic, Nelida Crnjaric-Zic, Zlatko Drmac, Aleksander Andrejcuk
Talk time: Thursday, May 25, 2:45 – 3:10 PM
Symposium: MS156 Applications of Koopman Operator Theory in Dynamical Systems:
Abstract: We propose new methods for the evaluation of eigenvalues of the Koopman operator family of the non-autonomous dynamic systems. The first step in the development is a new data-driven method for very accurate evaluation of eigenvalues in the hybrid linear non-autonomous case. Then, the approach is extended to continuous linear non-autonomous systems and non-autonomous systems in general. We also propose a relationship between eigenvalues and eigenvalues computed by Arnoldi-like methods on large sets of snapshots. We apply the new approach to baroreflex physiology, i.e. to the resonant breathing which is used in PTSD treatment. For the resonant breathing, we have both data and parameterized mathematical model. The model incorporates a delay and thus is infinite dimensional, hybrid, with a stochastic input. Its asymptotic dynamics is close to quasi-periodic. The applied new methods give an improved insight to the eigenvalues of the related Koopman operator family.