'We will use geometrical and analytical methods from wave physics to study exponentially small phenomena in dynamical systems. Many systems of interest in modern technology, from engineering to medicine, exhibit complex behaviours that arise from simple underlying laws -- a property called emergence. These can have huge consequences for a system's dynamics, causing instability and unpredictability. But the origin of such behaviour can be exponentially small, known as 'beyond all orders' because it lies beyond the precision achievable with standard analytical techniques. As such, many problems concerning whether emergent phenomena occur in important systems of interest remain unsolved. We will approach the problem from a fresh perspective, using mathematics such as stationary phase and Stokes phenomenon, first discovered in the 1840's, which have led to sophisticated ideas such as hyperasymptotics, discovered only recently. These methods are traditionally used to study wave problems in physics and provide insight into the geometry of exponentially small phenomena. They have been linked to the role of resurgence in dynamical systems, whereby exponentially small terms rise to prominence in emergent phenomena, but have not been applied systematically to these dynamical problems.
The emergent phenomena of interest arise when intersections are broken between special asymptotically attracting/repelling solutions, called stable and unstable manifolds. When the intersection breaks the manifolds split by an exponentially small amount. The problem is to calculate the splitting between manifolds to determine whether any intersection persists. A non-persistent intersection can imply the onset of chaotic dynamics, while intermittent intersections can imply the existence of canards -- trajectories that give rise to explosions of large amplitude `relaxation' oscillations. With our methods borrowed from physics we will calculate splittings that were previously incalculable.'