Autonomous choices among deterministic evolution-laws as source of uncertainty

TL;DR

This paper demonstrates that certain one-dimensional dynamical systems exhibit extreme sensitivity to initial conditions and choices of evolution rules, leading to intrinsic uncertainty in deterministic chaos, with hyperchaotic sequences expressible in closed form.

Contribution

It introduces a family of pseudo-random dynamical systems where the choice of evolution rule influences chaos, extending theories of complex behavior based on intrinsic uncertainty.

Findings

Hyperchaotic sequences can be expressed in closed form.

Sensitivity to initial conditions includes the choice of evolution rules.

The results suggest a new perspective on intrinsic uncertainty in deterministic systems.

Abstract

We provide evidence of an extreme form of sensitivity to initial conditions in a family of one-dimensional self-ruling dynamical systems. We prove that some hyperchaotic sequences are closed-form expressions of the orbits of these pseudo-random dynamical systems. Each chaotic system in this family exhibits a sensitivity to initial conditions that encompasses the sequence of choices of the evolution rule in some collection of maps. This opens a possibility to extend current theories of complex behaviors on the basis of intrinsic uncertainty in deterministic chaos.