Koopman operator and its approximations for systems with symmetries

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.