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 possesses a specific structure: aquadratic dissipative term coupled with a nonlinear restoring force that becomes singularat the left boundary and is modulated by a monotone weighting function. Under minimalregularity and monotonicity assumptions, we establish the existence, uniqueness, andglobal asymptotic stability of an equilibrium point. We also prove the structural stabilityof the attractor under small parameter perturbations. The key tool is an auxiliary functionH(x) whose geometric properties completely determine the system dynamics througha dimensionless parameter κ = β/α. Our results are purely mathematical, independent ofany physical interpretation, and provide a reusable foundation for applications in dynamicalsystems theory.