Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent statistics.

We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data.

Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization.

Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modelling high-dimensional systems from data. However, the quality of the linear DMD model is known to be fragile with respect to strong nonlinearity, which contaminates the model estimate.

Periodic Schwarz-Christoffel mappings with multiple boundaries per period.

We present an extension to the theory of Schwarz-Christoffel (S-C) mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal (straight-line) boundaries per period. Taking the arrangements to be periodic in the -direction in an (, )-plane, three cases are considered; these differ in whether the period window extends off to infinity as  →  ± ∞, or extends off to infinity in only one direction ( →  + ∞ or  →  - ∞), or is bounded. The preimage domain is taken to be a multiply connected circular domain.