We study a prototypical example in nonlinear dynamics where transition to self-similarity in a
singular limit is fundamentally changed as a parameter is varied. Here, we focus on the complicated
dynamics that occur in a generalised unstable thin-film equation that yields finite-time rupture. A
parameter, n, is introduced to model more general disjoining pressures. For the standard case of
van der Waals intermolecular forces, n = 3, it was previously established that a countably infinite
number of self-similar solutions exist leading to rupture. Each solution can be indexed by a parameter,
ϵ = ϵ1 > ϵ2 > · · · > 0, and the prediction of the discrete set of solutions requires examination of
terms beyond-all-orders in ϵ. However, recent numerical results have demonstrated the surprising
complexity that exists for general values of n. In particular, the bifurcation structure of self-similar
solutions now exhibits branch merging as n is varied. In this work, we shall present key ideas of how
branch merging can be interpreted via exponential asymptotics.