Spectral Thresholds in Dissipative Systems Under Finite-Spectrum Periodic Forcing

Students analyzing driven dissipative systems often correctly apply local dynamical equations while nevertheless drawing incorrect conclusions about global system behavior. This article identifies a recurrent reasoning error in the treatment of periodically forced dissipative systems: the implicit reduction of a finite-spectrum forcing to a single effective frequency. We show that this approximation leads to incorrect conclusions regarding resonance, stability, and state accessibility, even when all equations of motion are formally correct. By contrasting a plausible but flawed reasoning path with a correct spectral analysis, we clarify the role of spectral thresholds in dissipative systems. The discussion is intended for upper-undergraduate and early graduate courses in classical mechanics and dynamical systems, and is supported by original schematic figures and references to physics education research.