Basins of Attraction and Dynamic Integrity in 1-DOF Autonomous Systems with Quadratic Drag

Quadratic air drag is a standard dissipative mechanism in mechanical models, particularly when velocities are not small. In nonlinear systems, where multiple attractors may co-exist, analytical solutions for the motion are rarely available. For conservative one-degree-of-freedom systems, conservation of energy can be used to determine separatrices, dividing phase space into regions with qualitatively different dynamics, such as bounded and unbounded motion. However, in the presence of damping, energy conservation no longer holds, and analytical descriptions of basin-of-attraction boundaries remain scarce. This paper studies autonomous one-degree-of-freedom systems with air drag of the form $\ddot{x}+\alpha(x)|\dot{x}|\dot{x}+V'(x)=0,$ and targets the constructive determination of the \emph{basins of attraction} and the associated basin boundaries in the $(x,\dot{x})$ plane.On trajectory segments with fixed velocity sign, the substitution $p(x)=\dot{x}^2$ converts the second-order dynamics into a first-order linear equation for $p$ as a function of $x$. For the polynomial potentials considered here, this reduction yields explicit integral representations and, in several cases, closed-form expressions that can be continued branchwise from the saddle to generate separatrix-type curves in phase space. These curves provide exact boundaries that partition initial conditions either into different capture regimes or into capture versus escape, depending on the potential under investigation. In addition, the local integrity measures associated with the attractors are obtained analytically. Analytical results are verified against direct time-domain simulations. The resulting closed-form formulas provide benchmark cases for validating numerical algorithms that aim to determine dynamic integrity measures. Mathematics Subject Classification (2020) 34C15 · 34A34 · 37C29 · 37N15 · 70K44