Universal functions and exactly solvable chaotic systems

TL;DR

This paper explores the connection between universal functions and chaotic systems, demonstrating how solutions to certain chaotic systems can approximate any continuous function and exhibit noise-like behavior.

Contribution

It introduces a framework linking universal functions to chaotic systems, constructing solutions with noise-like properties and discussing their physical realizability.

Findings

Solutions to chaotic systems can approximate arbitrary continuous functions.

Universal functions can behave as delta-correlated noise.

Constructed universal dynamical systems exhibit truly noisy solutions.

Abstract

A universal differential equation is a nontrivial differential equation the solutions of which approximate to arbitrary accuracy any continuous function on any interval of the real line. On the other hand, there has been much interest in exactly solvable chaotic maps. An important problem is to generalize these results to continuous systems. Theoretical analysis would allow us to prove theorems about these systems and predict new phenomena. In the present paper we discuss the concept of universal functions and their relevance to the theory of universal differential equations. We present a connection between universal functions and solutions to chaotic systems. We will show the statistical independence between $X(t)$ and $X(t + \tau)$ (when $\tau$ is not equal to zero) and $X(t)$ is a solution to some chaotic systems. We will construct universal functions that behave as delta-correlated noise. We will construct universal dynamical systems with truly noisy solutions. We will discuss physically realizable dynamical systems with universal-like properties.