Universality of the route to chaos: Exact analysis

Abstract

The universality of the route to chaos is analytically proven for a countably infinite number of maps by proposing Super Generalized Boole (SGB) transformations. One of the routes to chaos, the intermittency route, was previously studied extensively by numerical methods. These researchers conjectured the universality in Type 1 intermittency, namely that the critical exponent of the Lyapunov exponent in this type of intermittency is 1/2. We prove their conjecture by showing that, for certain parameter ranges, the SGB transformations are exact and preserve the Cauchy distribution. Using the property of exactness, we prove that the critical exponent is 1/2 for a countably infinite number of maps where Type 1 intermittency occurs.