The simplest map with three-frequency quasi-periodicity and quasi-periodic bifurcations

TL;DR

This paper introduces a new three-dimensional map that exhibits two- and three-frequency quasi-periodicity, analyzing its bifurcations and comparing it with non-autonomous models using Lyapunov methods.

Contribution

A novel three-dimensional map demonstrating all basic quasi-periodic bifurcations, including features related to quasi-periodic Hopf bifurcation.

Findings

Demonstrates all basic quasi-periodic bifurcations in the map

Analyzes the three-parameter structure of bifurcations

Compares autonomous map behavior with non-autonomous models

Abstract

We propose a new three-dimensional map that demonstrates the two- and three-frequency quasi-periodicity. For this map all basic quasi-periodic bifurcations are possible. The study was realized by using method of Lyapunov charts completed by plots of Lyapunov exponents, phase portraits and bifurcation trees illustrating the quasi-periodic bifurcations. The features of the three-parameter structure associated with quasi-periodic Hopf bifurcation are discussed. The comparison with non-autonomous model is carried out.