We study the influence of decaying perturbations on autonomous oscillatory systems in a plane under the assumption that the perturbations preserve the equilibrium state of the limit system, oscillate with asymptotically constant frequency, and satisfy the nonresonance condition. We discuss the long-term behavior of the perturbed trajectories in a neighborhood of the equilibrium state. We describe conditions on the perturbation parameters that guarantee preservation or loss of stability of the equilibrium. The results are illustrated by an example of decaying perturbations of the Duffing oscillator.
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References
R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York etc. (1953).
L. D. Pustyl’nikov, “Stable and oscillating motions in nonautonomous dynamical systems. A generalization of C. L. Siegel’s theorem to the nonautonomous case,” Math. Sb. 23, No. 3, 382–404 (1974).
H. Thieme, “Asymptotically autonomous differential equations in the plane,” Rocky Mt. J. Math. 24, No. 1, 351–380 (1994).
J. A. Langa, J. C. Robinson, and A. Suárez, “Stability, instability and bifurcation phenomena in nonautonomous differential equations,” Nonlinearity 15, No. 3, 887–903 (2002).
P. E. Kloeden and S. Siegmund, “Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 3, 743–762 (2005).
M. Rasmussen, “Bifurcations of asymptotically autonomous differential equations,” Set-Valued Anal. 16, No. 7-8, 821–849 (2008).
L. Hatvani, “On the asymptotic stability for nonlinear oscillators with time-dependent damping,” Qual. Theory Dyn. Syst. 18, No. 2, 441–459 (2019).
N. N. Bogolyubov and Yu. A. Mitropol’skij, Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon and Breach, New York . (1961).
V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, Berlin (1997).
V. Burd, Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications, Chapman and Hall/CRC, Boca Raton, FL (2007).
J. D. Dollard and C. N. Friedman, “Existence of the Moller wave operators for \( V(r)=\frac{\gamma sin\left(\mu {r}^{\alpha}\right)}{r^{\beta }} \),” Ann. Phys. 111, 251–266 (1978).
A. Kiselev, “Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials,” Commun. Math. Phys. 179, No. 2, 377–399 (1996).
V. Burd and P. Nesterov, “Parametric resonance in adiabatic oscillators,” Result. Math. 58, No. 1–2, 1–15 (2010).
M. Lukic, “A class of Schrödinger operators with decaying oscillatory potentials,” Commun. Math. Phys. 326, No. 2, 441–458 (2014).
J. Brüning, S. Yu. Dobrokhotov, and M. A. Poteryakhin, “Averaging for Hamiltonian systems with one fast phase and small amplitudes,” Math. Notes 70, No. 5, 599–607 (2001).
S. Yu. Dobrokhotov and D. S. Minenkov, “On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation,” Regul. Chaotic Dyn. 15, No. 2–3, 285–299 (2010).
O. Sultanov, “Stability and asymptotic analysis of the autoresonant capture in oscillating systems with combined excitation,” SIAM J. Appl. Math. 78, No. 6, 3103–3118 (2018).
O. A. Sultanov, “Bifurcations of autoresonant modes in oscillating systems with combined excitation,” Stud. Appl. Math. 144, No. 2, 213–241 (2020).
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Translated from Problemy Matematicheskogo Analiza 111, 2021, pp. 137-149.
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Sultanov, O.A. Decaying Oscillatory Perturbations of Hamiltonian Systems in the Plane. J Math Sci 257, 705–719 (2021). https://doi.org/10.1007/s10958-021-05511-2
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DOI: https://doi.org/10.1007/s10958-021-05511-2