The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics

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=ϵ ₁>ϵ ₂>⋯>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.