TY - JOUR

T1 - Инвариантные торы периодических систем с девятью особенностями в гамильтоновой невозмущенной части

AU - Басов, Владимир Владимирович

AU - Жуков, Артем Сергеевич

PY - 2025

Y1 - 2025

N2 - The set of T -periodic systems of ODE with small parameter ε ≥ 0 ẋ = (γ(y3 − y) + Xν(t, x, y, ε)ε)εν, ẏ = (−(x3 − x) + Y ν(t, x, y, ε)ε)εν, (∗) where γ ∈ (0, 1], ν = 0, 1; Xν(t, x, y, 0), Y ν(t, x, y, 0) are real-analytic functions, and an unperturbed part, determined by the Hamiltonian H = (2x2 − x4 + γ(2y2 − y4))εν/4, has nine zeroes, is studied. For each zero of the Hamiltonian we explicitly nd the conditions on Xν(t, x, y, 0), Y ν(t, x, y, 0), 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 ε, neighbourhood of the cylindrical surface with the generating cycle as generatrix, for any small values of parameter, any 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" (ν = 1) and "slow" (ν = 0) 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 to the system (∗) 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 ε in its right-hand side (ν = 0), which describe the oscillations of the weakly-coupled oscillators.

AB - The set of T -periodic systems of ODE with small parameter ε ≥ 0 ẋ = (γ(y3 − y) + Xν(t, x, y, ε)ε)εν, ẏ = (−(x3 − x) + Y ν(t, x, y, ε)ε)εν, (∗) where γ ∈ (0, 1], ν = 0, 1; Xν(t, x, y, 0), Y ν(t, x, y, 0) are real-analytic functions, and an unperturbed part, determined by the Hamiltonian H = (2x2 − x4 + γ(2y2 − y4))εν/4, has nine zeroes, is studied. For each zero of the Hamiltonian we explicitly nd the conditions on Xν(t, x, y, 0), Y ν(t, x, y, 0), 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 ε, neighbourhood of the cylindrical surface with the generating cycle as generatrix, for any small values of parameter, any 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" (ν = 1) and "slow" (ν = 0) 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 to the system (∗) 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 ε in its right-hand side (ν = 0), which describe the oscillations of the weakly-coupled oscillators.

KW - Hamiltonian system

KW - Hilbert's cyclicity number

KW - averaging

KW - bifurcation

KW - invariant torus

KW - limit cycle

UR - https://www.mendeley.com/catalogue/2933f790-194d-396b-86c3-6150e8a7936f/

U2 - 10.21638/11701/spbu35.2025.107

DO - 10.21638/11701/spbu35.2025.107

M3 - статья

SP - 107

EP - 181

JO - ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ПРОЦЕССЫ УПРАВЛЕНИЯ

JF - ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ПРОЦЕССЫ УПРАВЛЕНИЯ

SN - 1817-2172

IS - 1

ER -