Crisis and unstable dimension variability in the bailout embedding map

TL;DR

This paper investigates the complex dynamics of inertial particles, specifically aerosols, in 2D flows using bailout embedding maps, revealing crises, intermittency, and unstable dimension variability near critical parameters.

Contribution

It provides the first detailed analysis of crisis phenomena and unstable dimension variability in aerosol dynamics modeled by bailout embedding maps.

Findings

Attractor widening and merging crisis observed in aerosol phase space.

Crisis-induced intermittency with power-law laminar length distribution (exponent -1/3).

Unstable dimension variability confirmed by Lyapunov exponent analysis.

Abstract

The dynamics of inertial particles in $2-d$ incompressible flows can be modeled by $4-d$ bailout embedding maps. The density of the inertial particles, relative to the density of the fluid, is a crucial parameter which controls the dynamical behaviour of the particles. We study here the dynamical behaviour of aerosols, i.e. particles heavier than the flow. An attractor widening and merging crisis is seen the phase space in the aerosol case. Crisis induced intermittency is seen in the time series and the laminar length distribution of times before bursts gives rise to a power law with the exponent $\beta=-1/3$. The maximum Lyapunov exponent near the crisis fluctuates around zero indicating unstable dimension variability (UDV) in the system. The presence of unstable dimension variability is confirmed by the behaviour of the probability distributions of the finite time Lyapunov exponents.