Bifurcations and degenerate periodic points in a three dimensional chaotic fluid flow

TL;DR

This paper investigates the role of degenerate periodic points in 3D chaotic fluid flows, revealing how bifurcations at these points influence local stability, transport, and the transition to chaos.

Contribution

It identifies and analyzes two types of degenerate periodic points in 3D fluid flows, detailing their impact on stability and transport, and provides conditions for tangent bifurcations in such systems.

Findings

Period-tripling bifurcations reverse local rotation angles and influence transport and manifold structures.

Tangent bifurcations create saddle-centre and period-doubling bifurcations, affecting chaos and stability.

Degenerate points significantly shape the global transport dynamics in 3D chaotic flows.

Abstract

Analysis of the periodic points of a conservative periodic dynamical system uncovers the basic kinematic structure of the transport dynamics, and identifies regions of local stability or chaos. While elliptic and hyperbolic points typically govern such behaviour in 3D systems, degenerate (parabolic) points play a more important role than expected. These points represent a bifurcation in local stability and Lagrangian topology. In this study we consider the ramifications of the two types of degenerate periodic points that occur in a model 3D fluid flow. (1) Period-tripling bifurcations occur when the local rotation angle associated with elliptic points is reversed, creating a reversal in the orientation of associated Lagrangian structures. Even though a single unstable point is created, the bifurcation in local stability has a large influence on local transport and the global arrangement of manifolds as the unstable degenerate point has three stable and three unstable directions, similar to hyperbolic points, and occurs at the intersection of three hyperbolic periodic lines. The presence of period-tripling bifurcation points indicates regions of either chaos or confinement depending on the nature of the associated manifold intersections. (2) The second type of bifurcation occurs when periodic lines become tangent to local or global invariant surfaces. This bifurcation creates both saddle-centre bifurcations which can create both chaotic and stable regions, and period-doubling bifurcations which are a common route to chaos in 2D systems. We provide conditions for the occurrence of these tangent bifurcations in 3D conservative systems, as well as constraints on the possible types of tangent bifurcation that can occur based on topological considerations.