Autoresonance in oscillating systems with combined excitation and weak dissipation

Sultanov, Oskar

doi:10.1016/j.physd.2020.132835

Cited by 8 publications

(9 citation statements)

References 29 publications

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“…Note that the series in ( 6) are assumed to be asymptotic as t → ∞ uniformly for all (x, y) ∈ B r (see, for example, [29, §1]). Such decaying perturbations appear, for example, in the study of Painlevé equations [30,31], resonance and phase-locking phenomena [32,33] and in many other problems associated with nonlinear and non-autonomous systems [34][35][36]. It can easily be checked that the rational powers of the form k/q with q > 1 in ( 6) can be reduced to the integer exponents k by the change of the time variable θ = t 1/q in system (1).…”

Section: Problem Statementmentioning

confidence: 99%

“…Define the function τ ε (t) = min{τ ε , t}, then z(τ ε (t)) is the process stopped at a first exit time from B ε . From (33) it follows that U (x(τ ε (t)), y(τ ε (t)), τ ε (t)) is a non-negative supermartingale (see, for example, [44, §5.2]). Hence, using (32) and the definition of τ ε (t), we get the following:…”

Section: Stability Analysis In Case (13)mentioning

confidence: 99%

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Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

Sultanov¹

2021

Preprint

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The effect of multiplicative stochastic perturbations on Hamiltonian systems on the plane is investigated. It is assumed that perturbations fade with time and preserve a stable equilibrium of the limiting system. The paper investigates bifurcations associated with changes in the stability of the equilibrium and with the appearance of new stochastically stable states in the perturbed system. It is shown that depending on the structure and the parameters of the decaying perturbations the equilibrium can remain stable or become unstable. In some intermediate cases, a practical stability of the equilibrium with estimates for the length of the stability interval is justified. The proposed stability analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions.

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Section: Problem Statementmentioning

confidence: 99%

Section: Stability Analysis In Case (13)mentioning

confidence: 99%

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

Sultanov¹

2021

Preprint

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“…Note that decreasing perturbations with power-law asymptotics appear, for example, in the study of Painlevé equations [26,27], phase-locking phenomena [28][29][30], stochastic perturbations [31,32], and in a wide range of other problems associated with nonlinear non-autonomous systems [33,34]. The evolution of I(t) = H(x(t), ẋ(t)) for solutions of ( 5) with different values of the parameters B, s and initial data.…”

Section: Problem Statementmentioning

confidence: 99%

Resonances in asymptotically autonomous systems with a decaying chirped-frequency excitation

Sultanov¹

2021

Preprint

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The influence of oscillatory perturbations on autonomous strongly nonlinear systems in the plane is investigated. It is assumed that the intensity of perturbations decays with time, and their frequency increases according to a power law. The long-term behaviour of perturbed trajectories is discussed. It is shown that, depending on the structure and the parameters of perturbations, there are at least two different asymptotic regimes: a phase locking and a phase drifting. In the case of phase locking, resonant solutions with an unlimitedly growing energy occur. The stability and asymptotics at infinity of such solutions are investigated. The proposed analysis is based on a combination of the averaging technique and the method of Lyapunov functions.

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“…Such perturbations appear, for example, in the study of solutions of Painlevé equations and their disturbances [9,10], in the asymptotic analysis of resonance phenomena in nonlinear systems [11][12][13][14], and in many other problems related to nonlinear non-autonomous systems [15][16][17][18][19].…”

Section: Problem Statementmentioning

confidence: 99%

“…Finally, consider the particular solutions x * (t), y * (t) having asymptotics (12) with x n = ν ± , where ν − < ν + are defined by (14). In this case,…”

Section: Stability Analysismentioning

confidence: 99%

Damped perturbations of systems with centre-saddle bifurcation

Sultanov

2020

Preprint

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An autonomous system of ordinary differential equations in the plane with a centre-saddle bifurcation is considered. The influence of time damped perturbations with power-law asymptotics is investigated. The particular solutions tending at infinity to the fixed points of the limiting system are considered. The stability of these solutions is analyzed when the bifurcation parameter of the unperturbed system takes critical and non-critical values. Conditions that ensure the persistence of the bifurcation in the perturbed system are described. When the bifurcation is broken, a pair of solutions tending to a degenerate fixed point of the limiting system appears in the critical case. It is shown that, depending on the structure and the parameters of the perturbations, one of these solutions can be stable, metastable or unstable, while the other solution is always unstable.

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Autoresonance in oscillating systems with combined excitation and weak dissipation

Cited by 8 publications

References 29 publications

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

Resonances in asymptotically autonomous systems with a decaying chirped-frequency excitation

Damped perturbations of systems with centre-saddle bifurcation

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