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In my problem I have non autonomous Hamiltonian which depends on 2 parameters (pretty close to oscillator Hamiltonian, $(a+b\cos t +1) p^2+(a+b\cos t-1)q^2$, $a,b$ - parameters). From numerical experiments and by some other ways its proven that for some values of parameters the monodromy matrix $M$ is identity - all solutions are periodic with period $2\pi$. Is there any kind of necessary condition for Hamiltonian so this could happen? For example its clear from numerical experiments that this happens only for integer values of $a$ what is pretty strange.

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"It is clear from numerical experiments ...": are you talking about evidence from numerical experiments, or rather computational verification? -- These are two different things ... . –  Stefan Kohl Feb 2 at 17:48
    
Have you tried to rewrite the equations of motion as Mathieu equations? –  Carl Feb 6 at 7:19

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