Nonlinear dynamic response of heavy particles in rotational vortices

Abstract The transport of inertial particles in vortical flows underpins a wide range of environmental processes, from the dispersion of microplastics in the ocean to aerosol clustering in the atmosphere. Yet, understanding their dynamics is limited due to the nonlinear interplay of drag, lift, buoyancy, and rotational forces. Here, we analyse particle motion in an analytically prescribed three-dimensional vortex flow, employing bifurcation theory and time series analysis to uncover the mechanism governing clustering, oscillations, and escape. We show that neutrally buoyant particles undergo a transition from stable to unstable equilibrium via a limit point bifurcation, while slightly negatively buoyant particles remain stable across a broad range of Stokes numbers. Inclusion of the Magnus lift is shown to be essential, as its omission conceals critical equilibrium branches and oscillatory states. Hopf bifurcation marks the onset of oscillatory motion, with codimension-two bifurcation analysis revealing a stability boundary separating supercritical and subcritical regimes by a Generalised Hopf bifurcation Point. In the supercritical case, particles exhibit pronounced axial oscillations at low Stokes numbers, reflecting the enhanced role of Magnus lift and vortex forcing. By contrast, the subcritical regime is restricted to a narrow vicinity in parameter space, where small perturbations trigger immediate divergence from equilibrium and particle escape. Further, the basin of attraction in the ( $$St-\bar{\rho }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>t</mml:mi> <mml:mo>-</mml:mo> <mml:mover> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> ) plane signifies the robustness of the bifurcation structure by showing the existence of three attractors in the stable region. These results provide a framework for predicting particle fate in vortical environments and extend to pollutant dispersion.