Related papers: A spectral characterization of nonlinear normal mo…
Nonlinear topology is an emerging field that combines the intrinsic reconfigurability of nonlinear systems with the robustness of topological protection, offering fertile ground for unconventional phenomena and novel applications. Recently,…
The Koopman-von Neumann equation describes the evolution of wavefunctions associated with autonomous ordinary differential equations and can be regarded as a quantum physics-inspired formulation of classical mechanics. The main advantage…
The emergence of Dynamic Mode Decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear PDE presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in…
We analyze the spectrum and normal mode representation of general quadratic bosonic forms $H$ not necessarily hermitian. It is shown that in the one-dimensional case such forms exhibit either an harmonic regime where both $H$ and…
Shape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the…
We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of the third order on a two dimensional manifold and show their application to the equivalence problem of such…
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal…
Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and…
The interplay between unitary dynamics and quantum measurements induces diverse phenomena in open quantum systems with no counterparts in closed quantum systems at equilibrium. Here, we generally classify Kraus operators and their effective…
The paper is about the computation of the principal spectrum of the Koopman operator (i.e., eigenvalues and eigenfunctions). The principal eigenfunctions of the Koopman operator are the ones with the corresponding eigenvalues equal to the…
Extended Dynamic Mode Decomposition (EDMD) is an algorithm that approximates the action of the Koopman operator on an $N$-dimensional subspace of the space of observables by sampling at $M$ points in the state space. Assuming that the…
Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant,…
An outstanding challenge in nonlinear systems theory is identification or learning of a given nonlinear system's Koopman operator directly from data or models. Advances in extended dynamic mode decomposition approaches and machine learning…
We provide one theorem of spectral equivalence of Koopman operators of an original dynamical system and its reconstructed one through the delay-embedding technique. The theorem is proved for measure-preserving maps (e.g. dynamics on compact…
Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This…
We consider a smooth submanifold $N$ with a smooth boundary in an ambient closed manifold $M$ and assign a spectral invariant $c(\alpha,H)$ to every singular homological class $\alpha\in H_*(N)$ and a Hamiltonian $H$ defined on the…
We study the structure of the normal matrix model (NMM). We show that all correlation functions of the model with axially symmetric potentials can be expressed in terms of holomorphic functions of one variable. This observation is used to…
Dynamic Mode Decomposition (DMD) is a technique to approximate generally non-linear dynamical systems using linear techniques, which are better understood and easier to analyze. Koopman theory extends DMD by transforming the original system…
Nonlinear optimal control is vital for numerous applications but remains challenging for unknown systems due to the difficulties in accurately modelling dynamics and handling computational demands, particularly in high-dimensional settings.…
For a compact connected Riemannian manifold with smooth boundary, we establish an effective procedure, by which we can calculate all the coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map associated to the…