A Comprehensive Analysis of Closedness of Solution Trajectories Around Singular Points in a Nonlinear Oscillator

The closedness of solution trajectories around singular points is a central question in the qualitative analysis of nonlinear oscillators, but it is rarely established rigorously when the restoring force has a finite-distance singularity. This work analyzes the single-degree-of-freedom oscillator u¨+u−λ/(a−u)=0, whose conserved first integral contains a logarithmic term, so that the local phase-portrait structure cannot be decided by linearization alone. The trajectories of the oscillator are determined directly from the first integral: the existence interval, monotonicity, extrema, and limiting values of the scalar functions appearing in the trajectory equation are established, and the level sets of the conserved quantity are then classified in every parameter regime. It is shown that, for λ<0, both singular points are centers surrounded by closed trajectories; for λ>0 with a2−4λ>0, one point is a center and the other a saddle, the level set through the saddle forming a separatrix; at a2−4λ=0, the coalesced point is a cusp admitting no closed trajectory; and for a2−4λ<0, no finite singular point exists. Numerical phase portraits generated from the first integral confirm the classification in every case. The analysis provides explicit conditions under which a center of the linearized system corresponds to actual closed nonlinear trajectories, thereby complementing the classical phase-plane treatment of nonlinear oscillations. The principal contribution of this work is the rigorous proof that the level sets of the first integral are closed, together with explicit existence, monotonicity, and limiting estimates for those level sets.