Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations

Oskar A. Sultanov

doi:10.3934/dcds.2021102

Article Contents

2021, Volume 41, Issue 12: 5943-5978. Doi: 10.3934/dcds.2021102 open access

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Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations

Oskar A. Sultanov

Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29, Saint Petersburg 199178, Russia

Received: September 30, 2020

Revised: December 31, 2020

Early access: June 2021

Published: December 2021

Research is supported by the Russian Science Foundation grant 19-71-30002

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Abstract

The effect of decaying oscillatory perturbations on autonomous Hamiltonian systems in the plane with a stable equilibrium is investigated. It is assumed that perturbations preserve the equilibrium and satisfy a resonance condition. The behaviour of the perturbed trajectories in the vicinity of the equilibrium is investigated. Depending on the structure of the perturbations, various asymptotic regimes at infinity in time are possible. In particular, a phase locking and a phase drifting can occur in the systems. The paper investigates the bifurcations associated with a change of Lyapunov stability of the equilibrium in both regimes. The proposed stability analysis is based on a combination of the averaging method and the construction of Lyapunov functions.

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Mathematics Subject Classification: Primary: 34C23, 34D20; Secondary: 37J65.

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Figure 1. The evolution of $r(t) = \sqrt{x^2(t)+y^2(t)}$ for solutions of (7) with $x(1) = 0.4$, $y(1) = 0$, $a = 4$, and $s_1 = 1$

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Figure 2. The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (47) with $h = 1/6$, $a_0 = a_1 = b_1 = s_2 = 0$, $s_1 = 1$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. Gray solid curves correspond to level lines of $H_0(x,y)$

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Figure 3. The evolution of $(x(t),y(t))$, $r(t)$, $\theta(t)$ for solutions of (47) with $h = 1/6$, $a_0 = a_1 = 0.8$, $b_1 = 0.6$, $s_1 = -0.8$, $s_2 = -1/6$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$. The gray dashed curve corresponds to $\theta = \psi_\ast$, where $\psi_\ast\approx -1.322$

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Figure 4. The evolution of $(x(t),y(t))$, $r(t)$, $\theta(t)$ for solutions of (47) with $h = 0$, $a_0 = 1$, $a_1 = 0.8$, $b_0 = 0$, $b_1 = 0.6$, $s_1 = -1$, $s_2 = -1/6$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue point corresponds to initial data $(x(1),y(1))$. The gray solid curve corresponds to $r = r(1) t^{{|\vartheta_4|}/{2}}$. The gray dashed curve corresponds to $\theta = \psi_\ast$, where $\psi_\ast\approx -1.322$

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Figure 5. The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (47) with $h = 1/6$, $a_0 = a_1 = 0.8$, $b_1 = 0.6$, $s_1 = -0.8$, $s_2 = 1$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$

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Figure 6. The evolution of $(x(t),y(t))$, $R(t)$, $\theta(t)$ for solutions of (50) with $h = 1/6$, $a_0 = a_1 = 0.8$, $b_1 = 0.6$, $s_1 = -1$, $s_2 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t)/2)$, $y(t) = -r(t)\sin(\theta(t)+S(t)/2)$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$. The gray dashed curves correspond to $\theta = \psi_\ast$, where $\psi_\ast\approx -1.4289$

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Figure 7. The evolution of $(x(t),y(t))$, $R(t)$, $|\theta(t)|$ for solutions of (50) with $h = 1/6$, $a_0 = a_1 = 0.8$, $b_1 = 0.6$, $s_1 = s_2 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t)/2)$, $y(t) = -r(t)\sin(\theta(t)+S(t)/2)$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$

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Figure 8. The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (51) with $h = 1/6$, $b_0 = -1/4$, $z_0 = 0.6$, $s_2 = 1$, $a_0 = a_1 = b_1 = z_1 = s_4 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue point corresponds to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$

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Figure 9. The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (51) with $h = 1/6$, $s_2 = 1$, $a_0 = a_1 = b_1 = z_1 = s_4 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. The gray solid curves correspond to level lines of $H_0(x,y)$. The gray dashed curve corresponds to $r = 2 R_\ast t^{-\nu-l/q}$

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Figure 10. The evolution of $(x(t),y(t))$, $r(t)$, $\theta(t)$ for solutions of (51) with $h = 0$, $a_0 = -1$, $a_1 = 1$, $s_2 = 1$, $s_4 = -1/4$, $b_1 = 0$, $\vartheta_8\approx -0.235$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$. The gray dashed curves correspond to $\theta = \psi_\ast$, where $\psi_\ast\approx -0.615$

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Figure 11. The evolution of $(x(t),y(t))$, $r(t)$, $|\theta(t)|$ for solutions of (51) with $h = 0$, $a_0 = -1$, $a_1 = 1$, $s_2 = 1$, $s_4 = 0$, where $x(t) = r(t)\cos(\theta(t)+S(t))$, $y(t) = -r(t)\sin(\theta(t)+S(t))$. The blue points correspond to initial data $(x(1),y(1))$

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