We know that the stability of Hamiltonian systems is at most neutral and so it can be violated by some non-Hamiltonian perturbation. However, if we confine ourselves to the space of Hamiltonian systems, then the notion of strong stability is meaningful. The stability of a linear Hamiltonian system is strong if it is preserved by any small deformation of the Hamiltonian.
In the Time-Invariant case, the (strong) stability of the Hamiltonian system is implied by the strong stability of its linearization. On the other hand, a linear periodically time-varying Hamiltonian system is equivalent to some linear time-invariant Hamiltonian system, by a linear time-varying transformation. Can we construct a local periodically time-varying nonlinear transformation to obtain a time-invariant Hamiltonian system from the periodically time-varying one? If so, similar to the time-invariant case, we can conclude stability properties of the periodically time-varying Hamiltonian system from its linearization.
I have not found an (explicit) theorem on it, in the well-known references.
@Salman
I would recommend "Mathematical Methods of Classical Mechanics" by V. I. Arnold. For example, I have a second edition of this book and on page 115. there is a theorem which (as it seems to me) replies to your question.
@Salman Farsi
Read Paul Trap paper by Bowen in PRL .
Hope you will get desired information.
B.Rath
Dear Prof. Vladimir Dananic
Thank you for the recommendation.
The book is very useful.
I read the mentioned theorem and it seems good as a starting point.
But, in fact, I can't conclude my conjecture from it (and even other theorems that I find in the book).
Dear Prof. Biswanath Rath
I can't find the paper you recommend.
May I ask you to specify the paper or give the link.
Thank you very much for your time.
If I have well understood your question, my answer is that the Barbashin-Krasovskii theorem and the Krasovskii theorem apply the second (direct) Lyapunov method - criteria to give sufficient condition for stability - to nonlinear dynamical systems varying periodically; in particular, to Hamiltonian systems. B-K and K theorems show the conditions by which, respectively, asymptotic stability and instability are obtained without resorting to the linear approximation.
Some references:
E.A. Barbashin, N.N. Krasovskii, "On the stability of motion
in the large", Dokl. Akad. Nauk. 86 (1952) 453–456.
J.P. LaSalle, S. Lefschetz, "Stability by Lyapunov’s Direct
Method", Academic Press, New York, 1961.
@Massimo Scalia
Dear Professor, thank you for paying attention.
In fact, the question is about the possible stability information can be obtained from the linearization of the periodically time-varying Hamiltonian systems (perhaps, similar to the time-invariant case).
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