Universality of the Route to Chaos -- Exact Analysis

TL;DR

This paper analytically proves the universality of the route to chaos for an infinite class of maps, confirming the critical exponent of 1/2 in Type 1 intermittency through exact analysis of SGB transformations.

Contribution

It introduces the Super Generalized Boole transformations and proves the universality of the route to chaos for these maps, confirming the critical exponent conjecture.

Findings

Proves the universality of the route to chaos for infinite maps.

Confirms the critical exponent of 1/2 in Type 1 intermittency.

Shows SGB transformations preserve the Cauchy distribution and are exact.

Abstract

The universality of the route to chaos is analytically proven for countably infinite number of maps by proposing the Super Generalized Boole (SGB) transformations. As one of the route to chaos, intermittency route was studied by Pomeau and Manneville numerically. They conjectured the universality in Type 1 intermittency, that the critical exponent of the Lyapunov exponent is $1/2$ in Type 1 intermittency. In order to prove their conjecture, we showed that for certain parameter ranges, the SGB transformations are exact and preserve the Cauchy distribution. Using the property of exactness, we proved that the critical exponent is $1/2$ for countably infinite number of maps where Type 1 intermittency occurs.