Clustering, Chaos and Crisis in a Bailout Embedding Map

TL;DR

This paper investigates the complex dynamics of inertial particles in two-dimensional flows using a four-dimensional dissipative bailout embedding map, revealing rich behaviors including bifurcations, crises, and intermittency with implications for impurity dynamics.

Contribution

It introduces a novel four-dimensional dissipative bailout embedding map to analyze inertial particle dynamics in 2D flows, highlighting bifurcation structures and crisis phenomena.

Findings

Identification of periodic, chaotic, and mixed regimes in the embedding map.

Observation of an attractor merging and widening crisis for aerosols.

Power law behavior of characteristic times before crises.

Abstract

We study the dynamics of inertial particles in two dimensional incompressible flows. The particle dynamics is modelled by four dimensional dissipative bailout embedding maps of the base flow which is represented by 2-d area preserving maps. The phase diagram of the embedded map is rich and interesting both in the aerosol regime, where the density of the particle is larger than that of the base flow, as well as the bubble regime, where the particle density is less than that of the base flow. The embedding map shows three types of dynamic behaviour, periodic orbits, chaotic structures and mixed regions. Thus, the embedding map can target periodic orbits as well as chaotic structures in both the aerosol and bubble regimes at certain values of the dissipation parameter. The bifurcation diagram of the 4-d map is useful for the identification of regimes where such structures can be found. An attractor merging and widening crisis is seen for a special region for the aerosols. At the crisis, two period-10 attractors merge and widen simultaneously into a single chaotic attractor. Crisis induced intermittency is seen at some points in the phase diagram. The characteristic times before bursts at the crisis show power law behaviour as functions of the dissipation parameter. Although the bifurcation diagram for the bubbles looks similar to that of aerosols, no such crisis regime is seen for the bubbles. Our results can have implications for the dynamics of impurities in diverse application contexts.