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

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 ̈ 𝑥 + 𝛼(𝑥 ) |̇𝑥 |̇ 𝑥 + 𝛽(𝑥 ) sgn(̇𝑥 ) + 𝑉 ′ (𝑥 ) = 0, and use a constructive alternative: on trajectory segments with fixed velocity sign, the squared-velocity substitution 𝑝(𝑥) =̇ 𝑥 2 converts the dynamics into a first-order linear equation for 𝑝 as a function of 𝑥. 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.
... mehr
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 ̇ 𝑥 = 0, 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
𝑈𝐹 (𝑥) = − cos 𝑥 − 𝐹𝑥, explicit separatrix formulae yield an escape-velocity criterion and a critical damping threshold separating unbounded transport from capture for |𝐹| < 1. 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

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 \[\ddot{x}+\alpha(x)\lvert\dot{x}\rvert\dot{x}+\beta(x)\operatorname{sgn}(\dot{x})+V'(x)=0,\] and use a constructive alternative: on trajectory segments with fixed velocity sign, the squared-velocity substitution \(p(x)=\dot{x}^{\,2}\) converts the dynamics into a first-order linear equation for \(p\) as a function of \(x\). ... mehrThis 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 \(\dot{x}=0\), 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 \(U_F(x)=-\cos x-Fx\), explicit separatrix formulae yield an escape-velocity criterion and a critical damping threshold separating unbounded transport from capture for \(\lvert F\rvert<1\). 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.