Diffusion-governed dynamical system for Coulombic rolling of cylinders bent by normal flux

This paper systematically analyzes a minimal differential–algebraic equation (DAE) model for the Coulombic rolling of flux-bent cylinders. Operating in the limit of negligible inertia, the system couples exponential curvature relaxation via diffusion with rotation-induced advection, constrained by a dry friction balance that mathematically links curvatures to rolling velocity. Based on an irreducible dimensionless system dependent on a single flux ratio q/q_c, I classify regimes of motion based on algebraic admissibility and physical viability, revealing inherent impasse singularities and parasitic solution branches. Bifurcation and stability analysis establishes a critical flux |q|=q_c (determined by rolling friction as well as cylinder properties) for a saddle–node bifurcation that yields stable and unstable steady rolling states jumping discontinuously from a stationary state. Furthermore, I identify a second threshold |q|/q_c=4/3^3/4 that fundamentally reconfigures the phase space topology. This second threshold reshapes the admissible manifolds, rewires branch connectivity, and flips the stability of fixed points by altering the reduced one-dimensional dynamics. Finally, I highlight the structural limitations of the disjoint constraint manifolds, emphasizing the need for future branch-transition mechanisms to fully resolve onset and cessation events.