My doubt is mostly regarding the time dependence. And if the term is also applicable to higher dimensions. For example, two copies of the previous periodically forced harmonic oscillator.
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Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Feb 23 '23 at 12:05
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Not really. There can be initial states where the solution is periodic with the period of the forcing, with others you get a mix of the eigenfrequency and the forcing frequency, or you get resonance. So you can get commensurable frequencies resulting in a periodic solution. Now if you add a dissipation/friction term to the DE, you can get something like a limit cycle based on the forcing with solutions converging to it. – Lutz Lehmann Feb 23 '23 at 16:59