Related papers: Linear operator theory of phase mixing
Analyzing synchronized nonlinear oscillators is one of the most important and attractive topics in nonlinear science. By understanding the interactions between the oscillators, we can figure out the synchronization process. A promising…
Koopman operator describes evolution of observables in the phase space, which could be used to extract characteristic dynamical features of a nonlinear system. Here, we show that it is possible to carry out interesting symbolic partitions…
We provide an overview of the Koopman operator analysis for a class of partial differential equations describing relaxation of the field variable to a stable stationary state. We introduce Koopman eigenfunctionals of the system and use the…
Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman…
The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function…
In this work, we study the existence of various classes of standing waves for a nonlinear Schr\"odinger system with quadratic interaction, along with a harmonic or partially harmonic potential. We establish the existence of ground-state…
The Koopman operator framework provides a perspective that non-linear dynamics can be described through the lens of linear operators acting on function spaces. As the framework naturally yields linear embedding models, there have been…
In this work, we propose to apply the recently developed Koopman operator techniques to explore the global phase space of a nonlinear system from time-series data. In particular, we address the problem of identifying various invariant…
Koopman analysis of a general dynamics system provides a linear Koopman operator and an embedded eigenfunction space, enabling the application of standard techniques from linear analysis. However, in practice, deriving exact operators and…
We consider Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals. The properties of the Koopman semigroup are described and a…
The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct…
The Koopman operator is a linear operator that describes the evolution of scalar observables (i.e., measurement functions of the states) in an infinitedimensional Hilbert space. This operator theoretic point of view lifts the dynamics of a…
Koopman operators and transfer operators represent nonlinear dynamics in state space through its induced action on linear spaces of observables and measures, respectively. This framework enables the use of linear operator theory for…
In this paper, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on SE(3) in terms of an infinite-dimensional linear system which evolves in the space of observable functions (lifted space) and…
Spectral decomposition of dynamical systems is a popular methodology to investigate the fundamental qualitative and quantitative properties of these systems and their solutions. In this chapter, we consider a class of nonlinear cooperative…
We provide a framework for learning of dynamical systems rooted in the concept of representations and Koopman operators. The interplay between the two leads to the full description of systems that can be represented linearly in a finite…
A number of physically intuitive results for the calculation of multi-time correlations in phase-space representations of quantum mechanics are obtained. They relate time-dependent stochastic samples to multi-time observables, and rely on…
Recently, Koopman operator theory has become a powerful tool for developing linear representations of non-linear dynamical systems. However, existing data-driven applications of Koopman operator theory, including both traditional and deep…
We investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain…
We provide a linear analysis on normal modes of the spin Boltzmann equation proposed in \cite{Weickgenannt:2021cuo}, where the non-diagonal or polarized part of the transition rate is neglected to ensure the Hermitian property of linearized…