On a Global Attractor for a Class of One-Dimensional Autonomous Systems

We study a class of one-dimensional autonomous ordinary differential equations definedon a bounded interval (0, xc) ⊂ (0, 1). The vector field combines a quadratic dissipativeterm with a nonlinear restoring force that becomes singular at the left boundary and ismodulated by a monotone weighting function. Under minimal regularity and monotonicityassumptions, we establish existence of equilibria and uniqueness of a stable equilibriumon the increasing branch of an auxiliary function H(x), together with its globalasymptotic stability within the invariant interval in the admissible regime.The dimensionless parameter κ = β/α controls the attractor location, with a computablecritical value κc given by the maximum of H. We also prove structural stability of thestable attractor under small parameter perturbations. The results are purely mathematicaland provide a reusable foundation for applications in dynamical systems theory.