Related papers: Approximate Dynamical Systems
This paper presents a study of the Koopman operator theory and its application to optimal control of a multi-robot system. The Koopman operator, while operating on a set of observation functions of the state vector of a nonlinear system,…
This article establishes the foundation for a new theory of invariant/integral manifolds for non-autonomous dynamical systems. Current rigorous support for dimensional reduction modelling of slow-fast systems is limited by the rare events…
We present a pedagogical work-in-progress. This textbook aims to introduce Hilbert space representations for quantum and classical dynamics, outlining the mathematical foundations, practical guidance, and Python implementation of dynamical…
We consider the basic features of complex dynamical and control systems. Special attention is paid to the problems of synthesis of dynamical models of complex systems, construction of efficient control models, and to the development of…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
Differentiable quantum dynamics require automatic differentiation of a complex-valued initial value problem, which numerically integrates a system of ordinary differential equations from a specified initial condition, as well as the…
We obtain exact analytic expressions for a class of functions expressed as integrals over the Haar measure of the unitary group in d dimensions. Based on these general mathematical results, we investigate generic dynamical properties of…
We analyze the quantization of dynamical systems that do not involve any background notion of space and time. We give a set of conditions for the introduction of an intrinsic time in quantum mechanics. We show that these conditions are a…
Given a finite-dimensional time continuous control system and $\varepsilon>0$, we address the question of the existence of controls that maintain the corresponding state trajectories in the $\varepsilon$-neighborhood of any prescribed path…
A functional differential equation related to the logistic equation is studied by a combination of numerical and perturbation methods. Parameter regions are identified where the solution to the nonlinear problem is approximated well by…
It is demonstrated that nonlinear dynamical systems with analytic nonlinearities can be brought down to the abstract Schr\"odinger equation in Hilbert space with boson Hamiltonian. The Fourier coefficients of the expansion of solutions to…
Starting form a microscopic system-environment model, we construct a quantum dynamical semigroup for the reduced evolution of the open system. The difference between the true system dynamics and its approximation by the semigroup has the…
The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The…
We develop a generalized hyperdynamics method, which is able to simulate slow dynamics in atomistic general (both energy and entropy-dominated) systems. We show that a few functionals of the pair correlation function, involving two-body…
Koopman analysis provides a general framework from which to analyze a nonlinear dynamical system in terms of a linear operator acting on an infinite-dimensional observable space. This theoretical framework provides a rigorous underpinning…
We study discrete-time $(m,n)$-multirate systems on separable Hilbert spaces, solving the problem of approximating such a system by one which has a shorter multirate period $(m/q,n/q)$, optimally in the Hilbert-Schmidt norm. We work in the…
A nonlinear operator equation $F(x)=0$, $F:H\to H,$ in a Hilbert space is considered. Continuous Newton's-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point $x_0$ and becoming…
We develop an approximation theory in Hilbert spaces that generalizes the classical theory of approximation by entire functions of exponential type. The results advance harmonic analysis on manifolds and graphs, thus facilitating data…
The dynamics of a wide range of technologically important quantum systems are dominated by their interaction with just a few environmental modes. Such highly structured environments give rise to long-lived bath correlations that induce…
We present a theory of compatible differential constraints of a hydrodynamic hierarchy of infinite-dimensional systems. It provides a convenient point of view for studying and formulating integrability properties and it reveals some hidden…