Basins of Attraction and Escape in Pendulum‐Type Potentials with Quadratic Air Drag

ABSTRACT

Quadratic air drag is a common dissipative mechanism at moderate to high velocities, yet in nonlinear one‐degree‐of‐freedom systems, it typically blocks closed‐form solutions and energy‐based phase‐plane partitions. We study the autonomous class and use a constructive alternative: on trajectory segments with fixed velocity sign, the squared‐velocity substitution converts the dynamics into a first‐order linear equation for as a function of . This yields explicit phase‐plane branches that can be continued and matched across turning points, providing separatrix‐type boundaries even when an explicit time parametrization is unavailable.

For the cosine potential, we derive analytic saddle‐separatrix branches and provide the critical‐velocity thresholds, as well as the winding‐number classification of rotations corresponding to each separatrix. Adding dry (Coulomb) friction introduces set‐valued dynamics at , producing sticking intervals and ribbon‐shaped regions of the initial conditions' plane leading to sticking; their boundaries follow from the same branchwise construction for constant, spatially varying, and normal‐force‐dependent friction laws. For the tilted periodic (washboard) potential , explicit separatrix formulae yield an escape‐velocity criterion and a critical damping threshold separating unbounded transport from capture for . Analytical boundaries and thresholds are validated against direct time‐domain simulations, and the resulting closed and semi‐closed expressions provide benchmark cases for numerical basin‐boundary and dynamical integrity computations in periodic settings.