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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

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Koopman operator and its approximations for systems with symmetries.

Anastasiya Salova¹, Jeffrey Emenheiser¹, Adam Rupe¹

¹Department of Physics, University of California, Davis, California 95616, USA.

Chaos (Woodbury, N.Y.)

|October 3, 2019

Summary

Symmetries in nonlinear systems simplify analysis using the Koopman operator. Incorporating symmetry into extended and kernel dynamic mode decomposition (EDMD and kernel DMD) methods reveals hidden organization and improves computation of spectral properties.

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Area of Science:

Nonlinear Dynamics
Dynamical Systems Theory
Applied Mathematics

Background:

Nonlinear dynamical systems with symmetries display complex behaviors like bifurcations and attractor-basin portraits.
Symmetry arguments are crucial for simplifying the analysis of these collective behaviors.
The Koopman operator offers a linear framework to represent nonlinear dynamics, preserving global features unlike local linearization.

Purpose of the Study:

To demonstrate how symmetries influence the Koopman operator's structure and spectral properties.
To show that symmetry considerations simplify Koopman operator approximation using extended and kernel dynamic mode decomposition (EDMD and kernel DMD).
To explore the impact of measurement noise on these methods.

Main Methods:

Utilized representation theory to analyze the effect of symmetries on the Koopman operator.
Applied extended and kernel dynamic mode decomposition (EDMD and kernel DMD) methods.
Investigated the block diagonal structure induced by isotypic component bases in operator approximations.

Main Results:

Symmetries induce a block diagonal structure in Koopman operator approximations when using an isotypic component basis.
This structure reveals inherent organization within the system's dynamics.
Symmetry-informed EDMD and kernel DMD enable more efficient computation of the Koopman operator approximation and its spectral properties (eigenvalues, eigenfunctions, eigenmodes).

Conclusions:

Symmetry considerations significantly simplify the analysis of nonlinear dynamical systems via the Koopman operator.
Modified EDMD and kernel DMD methods offer computational advantages when system symmetries are known.
The framework provides a pathway to better understand and compute spectral properties of symmetric nonlinear systems.