For the generalized Duffing oscillator, the origin is a global center (all orbits closed) if and only if damping is zero and the restoring force has a single-well potential with odd polynomial degree.
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Global Centers and Phase Portraits in Generalized Duffing Oscillators: A Comprehensive Study of the Center-Focus Problem
For the generalized Duffing oscillator, the origin is a global center (all orbits closed) if and only if damping is zero and the restoring force has a single-well potential with odd polynomial degree.