Maxwell Points of Dynamical Control Systems Based on Vertical Rolling Disc—Numerical Solutions

We study two nilpotent affine control systems derived from the dynamic and control of a vertical rolling disc that is a simplification of a differential drive wheeled mobile robot. For both systems, their controllable Lie algebras are calculated and optimal control problems are formulated, and their Hamiltonian systems of ODEs are derived using the Pontryagin maximum principle. These optimal control problems completely determine the energetically optimal trajectories between two states. Then, a novel numerical algorithm based on optimisation for finding the Maxwell points is presented and tested on these control systems. The results show that the use of such numerical methods can be beneficial in cases where common analytical approaches fail or are impractical.

Homoclinic snaking near the surface instability of a polarisable fluid

We report on localised patches of cellular hexagons observed on the surface of a magnetic fluid in a vertical magnetic field. These patches are spontaneously generated by jumping into the neighbourhood of the unstable branch of the domain-covering hexagons of the Rosensweig instability upon which the patches equilibrate and stabilise. They are found to coexist in intervals of the applied magnetic field strength parameter around this branch. We formulate a general energy functional for the system and a corresponding Hamiltonian that provide a pattern selection principle allowing us to compute Maxwell points (where the energy of a single hexagon cell lies in the same Hamiltonian level set as the flat state) for general magnetic permeabilities. Using numerical continuation techniques, we investigate the existence of localised hexagons in the Young–Laplace equation coupled to the Maxwell equations. We find that cellular hexagons possess a Maxwell point, providing an energetic explanation for the multitude of measured hexagon patches. Furthermore, it is found that planar hexagon fronts and hexagon patches undergo homoclinic snaking, corroborating the experimentally detected intervals. Besides making a contribution to the specific area of ferrofluids, our work paves the ground for a deeper understanding of homoclinic snaking of two-dimensional localised patches of cellular patterns in many physical systems.