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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Invariant Tori of the Periodic Systems with the Nine Equilibrium Points in Hamiltonian Unperturbed Part

Author(s):

Vladimir Vladimirovich Basov

St. Petersburg State University, Department of Differential Equations,
Candidate of Physics and Mathematics, Associate Professor

vlvlbasov@rambler.ru

Artem Sergeevich Zhukov

The Laboratory of Continuous Mathematical Education, senior methodologist

Abstract:

In this papaer we study the set of periodic with respect to T systems of ODE with small parameter e>0. An unperturbed part of each system is determined by special Hamiltonian, dependant on positive, not greater than 1 parameter, and has nine zeroes. Perturbations are real-analytic functions. For each zero of the Hamiltonian we explicitly find the conditions on perturbations, which allow to distinguish the sets of initial values for the initial value problem of the unperturbed system. These initial values parametrize so-called generating cycles. It is proven that in small, with respect to e, neighbourhood of the cylindrical surface with the generating cycle as generatrix, for any small values of parameter, any studied system has a two-periodic invariant surface, homeomorphic to torus, if time is factored with the respect to the period. Formulas and asymptotic expansions of this surface are provided, the number of properties is discovered. Original example of a set of systems with both "fast" and "slow" time is constructed. The perturbation, independent of parameter, in these systems have polynomial of the third degree with three terms as an average with the respect to $t.$ It was established that these systems have eleven invariant tori. Aforementioned results are obtained using a generating tori splitting method (GTS method). Detailed description of the algorithm of this method and the demonstration of its application are the second purpose of this paper. Developed method of searching for the invariant tori that remain for each small parameter value is universal, because it can be applied to the systems with an unperturbed parts, determined by any Hamiltonian, provided that the equilibrium points and separatrices of those systems can be found. GTS method, in particular, is an alternative to the so called detection functions method and Melnikov function method, which are used in studies concerning the weakened XVI Hilbert's problem on the evaluation of a number of limit cycles of autonomous systems with the hamiltonian unperturbed part. Thus, the GTS method allows to evaluate the lower bound of the analogue of the Hilbert's cyclicity value, which determines the amount of the invariant tori in the periodic systems with "slow" and "fast" time and different hamiltonian unperturbed parts. It is also used in the case of the periodic systems of any even degree with the common factor e in its right-hand side, which describes the oscillations of the weakly-coupled oscillators

Keywords

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