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Approximate research of problems on perturbation of periodic and autonomous Hamiltonian systems in critical cases. (English) Zbl 1458.37064

Authors’ abstract: The paper studies the problem of approximate construction of eigenvalues and multipliers of linear (autonomous and periodic) Hamiltonian systems depending on a small parameter in the main critical cases. New formulas for the asymptotic (in powers of a small parameter) representation of the eigenvalues and multipliers of the task are proposed. The obtained formulas allow us to effectively study the problems of stability and hyperbolicity of linear systems, equilibrium points and periodic solutions of nonlinear Hamiltonian systems, the problem of constructing the boundaries of the regions of stability and hyperbolicity, problems of local bifurcations of nonlinear dynamical systems, etc.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J46 Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
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References:

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