Abstract
The effect of multi-frequency quasi-periodic perturbations on systems close to twodimensional nonlinear Hamiltonian ones is studied. It is assumed that the corresponding perturbed autonomous system has a double limit cycle. Analysis of the Poincar´e–Pontryagin function constructed for the autonomous system makes it possible to establish the presence of such a cycle. When the condition of commensurability of the natural frequency of the corresponding unperturbed Hamiltonian system with the frequencies of the quasi-periodic perturbation is fulfilled, the unperturbed level becomes resonant. Resonant structures essentially depend on whether the selected resonance levels coincide with the levels that generate limit cycles in the autonomous system. An averaged system is obtained that describes the topology of the neighborhoods of resonance levels. Possible phase portraits of the averaged system are established near the bifurcation case, when the resonance level coincides with the level in whose neighborhood the corresponding autonomous system has a double limit cycle. To illustrate the results obtained, the results of a theoretical study and of a numerical calculation are presented for a specific pendulum-type equation under two-frequency quasi-periodic perturbations.
Highlights
Analysis of the Poincare–Pontryagin function constructed for the autonomous system makes it possible to establish the presence of such a cycle
Resonant structures essentially depend on whether the selected resonance levels coincide with the levels that generate limit cycles in the autonomous system
An averaged system is obtained that describes the topology of the neighborhoods of resonance levels
Summary
Предполагается, что собственная частота ω(I) невозмущенной системы является монотонной функцией и не обращается в нуль на интервале (Imin, Imax) ≡ (I(hmin), I(hmax)). Функции F1 и G1 – достаточно гладкие по переменным I, θ, θi, i = 1, p в области [Imin, Imax] × Tp+1, где Tp+1 – (p+1)-мерный тор. Где F2(I, θ) ≡ f0(X, Y )Xθ′ − g0(X, Y )Yθ′, G2(I, θ) ≡ −f0(X, Y )XI′ + g0(X, Y )YI′. В данной статье будем рассматривать случай, когда автономная система имеет двойной предельный цикл. Двукратный корень u = u∗ уравнения B0(u) = 0 (B1(u∗) = 0, B1′ (u∗) ̸= 0) определяет невозмущенный уровень I = I∗ (замкнутую фазовую кривую h = h∗ невозмущенной системы), от которого под действием возмущения родится двойной предельный цикл. Под действием неавтономного квазипериодического возмущения резонансный уровень может совпасть с уровнем невозмущенной системы, порождающим двойной предельный цикл в автономной системе. Настоящее исследование следует работам [9,10], [14]
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