Related papers: Universal functions and exactly solvable chaotic s…
We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a re duction of the associated dynamical…
The study of chaotic systems, where rare events play a pivotal role, is essential for understanding complex dynamics due to their sensitivity to initial conditions. Recently, tools from large deviation theory, typically applied in the…
We investigate universal behavior of isolated many-body systems far from equilibrium, which is relevant for a wide range of applications from ultracold quantum gases to high-energy particle physics. The universality is based on the…
An universal form of kinetic equation for open systems is considered which naturally unifies classical and quantum cases and allows to extend concept of wave function to open quantum systems. Corresponding stochastic Schr\"{o}dinger…
We establish fundamental mathematical limits on universal approximation theorem (UAT) system alignment by proving that catastrophic failures are an inescapable feature of any useful computational system. Our central thesis is that for any…
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations)…
Under certain general conditions, an explicit formula to compute the greatest delta-epsilon function of a continuous function is given. From this formula, a new way to analyze the uniform continuity of a continuous function is given.…
For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise,…
In this paper, a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite- and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further…
This paper concerns exact differential equations. First, I define two types of functions which I have named Basic Function of Type One and Basic Function of Type Two. I then derive the property and theorems of these functions. Finally, by…
The paper is focused on the discussion of the phenomenon of transitional chaos in dynamic autonomous and non-autonomous systems. This phenomenon involves the disappearance of chaotic oscillations in specific time periods and the system…
Chaotic systems exhibit rich quantum dynamical behaviors ranging from dynamical localization to normal diffusion to ballistic motion. Dynamical localization and normal diffusion simulate electron motion in an impure crystal with a vanishing…
We report on the possibilities of using the method of normal fundamental systems for solving some problems of oscillation theory. Large elastic dynamical systems with continuous and discrete parameters are considered, which have many…
In this paper, we first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a…
A multi-linear variable separation approach is developed to solve a differential-difference Toda equation. The semi-discrete form of the continuous universal formula is found for a suitable potential of the differential-difference Toda…
We introduce a new analytical method, which allows to find out chaotic dynamics in non-smooth dynamical systems. A simple mechanical system consisting of a mass and a dry friction element is considered as an example. The corresponding…
We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes,…
Generalized synchronization of chaos is a type of cooperative behavior in directionally-coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic…
We propose the existence of a new universality in classical chaotic systems when the number of degrees of freedom is large: the statistical property of the Lyapunov spectrum is described by Random Matrix Theory. We demonstrate it by…
Upper semicontinuous (usc) functions arise in the analysis of maximization problems, distributionally robust optimization, and function identification, which includes many problems of nonparametric statistics. We establish that every usc…