Bifurcation points arise in many problems involving interfacial flows, typically due to changes in a control parameter of the system, leading to transitions between dynamical regimes. Hence, understanding such phenomena is essential for gaining control over the behaviour of the system. In this talk, I will present recent advances on two different problems: the motion of droplets on patterned surfaces and pulse interactions in falling liquid films. I will show that in both scenarios, a hierarchy of bifurcations leads to a variety of different dynamics.

In the droplets problem, we observe the onset of pitchfork bifurcations as the volume of the droplet slowly changes over time. We demonstrate that at each bifurcation point, the droplet exhibits a rapid change in its position, which we quantify in terms of its instantaneous velocity. We find that there are two distinct dynamical regimes: an over-damped regime, dominated by viscous dissipation and surface tension, and an under-damped oscillatory regime, where inertial effects become significant. In the falling film problem, I will show that the interactions between two pulses strongly depend on the Reynolds number. As this parameter increases, we observe an unusual Hopf bifurcation that leads to a state of self-sustained oscillations. This dynamical state persists at higher Reynolds numbers until it loses stability, forcing the pulses to separate from each other.