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Universal functions and exactly solvable chaotic systems | | M. A. Garcia-Nustes
; Emilio Hernandez-Garcia
; Jorge A. Gonzalez
; | | Date: |
7 Nov 2008 | | 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
$X(t + au)$ (when $ au$ 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. | | Source: | arXiv, 0811.1179 | | Services: | Forum | Review | PDF | Favorites | |
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