Discrete map reduction of bursting flows in periodically driven cylindrical cavities

Highly complex flow patterns arising in a viscous fluid confined within a cylindrical cavity, driven by time-periodically oscillating walls, are studied numerically. To enable predictive analysis of these dynamics, we present a reduction based on a mathematical discrete map, formulated in the vicinity of a codimension-2 bifurcation. This map, derived as a normal form using equivariant bifurcation theory and scaled according to the physical parameters of the system, captures the essential features of the burst dynamics. The reduced model accurately reflects the core dynamics observed in Direct Numerical Simulations (dns). After appropriate scaling, the map predictions show excellent agreement with dns results, successfully forecasting the range of accessible flow solutions. Furthermore, the map reveals a coherent flow pattern with non-trivial time dependence, previously overlooked, offering new insights into its emergence. These insights clarify the origin of the competition between pulsating and bursting flows, showing that it arises from heteroclinic connections between quasiperiodic flows. The scaling laws corresponding to such collision have been confirmed after a careful swipe in the parameter space, involving the computation of an extensive number of bursting regimes.