Global dynamics of generalized Duffing oscillators with global centers

This study provides a comprehensive investigation of the generalized Duffing oscillatorx˙=y,y˙=−αy−ϵxm−σx,where α,σ∈R, ϵ ≠ 0 and m ≥ 1. We establish a complete topological classification of the phase portraits for all m ≥ 1 and all real parameter values. A key finding is the existence of global centers when the linear damping vanishes (α=0), with explicit conditions given for both linear (m=1) and nonlinear (m > 1) cases. To achieve this classification, we employ quasi-homogeneous blow-up techniques adapted to arbitrary degree m, overcoming the limitations of classical directional blow-up methods that typically require case-by-case analysis. In the conservative limit α=0, we fully resolve the center–focus problem and characterize the rich structure of homoclinic and heteroclinic cycles that organize the phase space. These results provide a unified framework for understanding the global dynamics of Duffing-type oscillators and identify the precise parameter regimes where purely periodic, conservative behavior emerges.