Abstracts

Yoshihiko Susuki, Department of Electrical Engineering, Kyoto University, Japan

Title: Koopman resolvents in dynamical systems and control

Abstract: The Koopman operator framework provides a new direction to the analysis and synthesis of systems with complex dynamics. In this talk, I present a comprehensive review of exploiting resolvent operators for Koopman operators and Koopman generators, also known as Koopman resolvents. The resolvent operator of a linear (bounded and unbounded) operator is a standard concept not only in functional analysis but also in mathematical physics. Here, rather than the operator-theoretic viewpoint, I introduce the dynamical-system viewpoint to the Koopman resolvents, demonstrating how the resolvent operators are applied to analyze properties of nonlinear systems. Here, I introduce nonlinear generalizations of the Laplace analysis and the frequency response, as well as numerical methods for nonlinear time-series analysis.

Alexandre Mauroy, Department of Mathematics & Namur Institute for Complex Systems (naXys), University of Namur, Belgium

Title: Analytic EDMD method for spectral analysis of fixed point dynamics

Abstract: We will present an EDMD-type method that captures the spectrum and eigenfunctions of the Koopman operator defined on a reproducing kernel Hilbert space of analytic functions. This approach relies on an orthogonal projection on polynomial subspaces, which is equivalent to data-driven Taylor approximation. In the case of dynamics with a hyperbolic equilibrium, the proposed method demonstrates excellent performance to capture the lattice structured Koopman spectrum, including the eigenvalues of the linearized system at the equilibrium. In particular, it remains efficient with partial state measurements or with a dataset generated far from the equilibrium. More importantly, this technique preserves and exploits the triangular structure of the operator so that it does not suffer from spectral pollution, reaching arbitrary accuracy on the spectrum with a fixed finite dimension of the approximation. This property will be demonstrated by numerical simulations and validated with theoretical error bounds. The effect of the kernel choice will also be investigated. Finally, applications of the method will be considered in specific settings such as spectral network identification, data-driven stability analysis, and prediction.

Hiroya Nakao, Department of Systems and Control Engineering, Institute of Science Tokyo, Japan

Title: Koopman operator analysis of coupled oscillator systems

Abstract: TBD

Yuzuru Kato, Department of Complex and Intelligent Systems, Future University Hakodate, Japan

Title: Analysis of quantum nonlinear oscillators on the basis of Koopman operator theory

Abstract: Synchronization of quantum nonlinear oscillators has recently attracted significant attention. In classical nonlinear dynamics, the asymptotic phase and asymptotic amplitude, fundamental quantities closely connected to the Koopman operator theory, provide powerful tools for analyzing limit-cycle oscillations and their transient behaviors. These notions have also been extended to stochastic oscillators, where the asymptotic phase and amplitude are defined using eigenfunctions of the backward Fokker–Planck (Kolmogorov) operator, which is an infinitesimal generator of the Koopman operator for the stochastic dynamics. In this talk, inspired by these stochastic asymptotic phases and amplitudes, we introduce a fully quantum-mechanical definition of the asymptotic phase and amplitude for quantum nonlinear oscillators. The proposed quantum asymptotic phase and amplitude are introduced using the eigenoperators of the adjoint Liouville superoperator, which serves as the infinitesimal generator of the Koopman operator for open quantum systems. Using this framework, we analyze both quantum limit-cycle oscillations and quantum noise-induced oscillations, demonstrating how the quantum asymptotic phases and amplitudes characterize their behavior.