Digit replacement: A generic map for nonlinear dynamical systems

TL;DR

This paper introduces a simple, discontinuous digit manipulation map as a versatile model for nonlinear dynamical systems, capable of generating various oscillations and complex attractors with exact solutions.

Contribution

It presents a novel, mathematically designed map using digit manipulation that models complex nonlinear dynamics and allows precise control over system behaviors.

Findings

Exact solutions for the orbit in wide parameter regions

Ability to generate regular and aperiodic oscillations

Construction of complex attractors with specific properties

Abstract

A simple discontinuous map is proposed as a generic model for nonlinear dynamical systems. The orbit of the map admits exact solutions for wide regions in parameter space and the method employed (digit manipulation) allows the mathematical design of useful signals, such as regular or aperiodic oscillations with specific waveforms, the construction of complex attractors with nontrivial properties as well as the coexistence of different basins of attraction in phase space with different qualitative properties. A detailed analysis of the dynamical behavior of the map suggests how the latter can be used in the modeling of complex nonlinear dynamics including, e.g., aperiodic nonchaotic attractors and the hierarchical deposition of grains of different sizes on a surface.