Bifurcations of autoresonant modes in oscillating systems with combined excitation

Sultanov, Oskar

doi:10.1111/sapm.12294

Cited by 10 publications

(7 citation statements)

References 31 publications

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“…The proposed method of study of asymptotic regimes in system (1) is based on the construction of appropriate Lyapunov functions. Recently, it was noted in [19][20][21] that such functions are effective in the asymptotic analysis of solutions to nonlinear non-autonomous systems. See also [22] for application of the second Lyapunov method to asymptotic analysis of equations with a small parameter.…”

Section: Change Of Variablesmentioning

confidence: 99%

Stability and bifurcation phenomena in asymptotically Hamiltonian systems

Sultanov

2022

Nonlinearity

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The influence of time-dependent perturbations on an autonomous Hamiltonian system with an equilibrium of center type is considered. It is assumed that the perturbations decay at infinity in time and vanish at the equilibrium. In this case the stability and the long-term behaviour of trajectories depend on nonlinear and non-autonomous terms of equations. The paper investigates bifurcations associated with a change of Lyapunov stability of the equilibrium and the emergence of new attracting or repelling states in the perturbed non-autonomous system. The dependence of bifurcations on the structure of perturbations is discussed.

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Section: Change Of Variablesmentioning

confidence: 99%

Stability and bifurcation phenomena in asymptotically Hamiltonian systems

Sultanov

2022

Nonlinearity

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“…The proposed method of study of asymptotic regimes in system ( 1) is based on the construction of appropriate Lyapunov functions. Recently, it was noted in [19][20][21] that such functions are effective in the asymptotic analysis of solutions to nonlinear non-autonomous systems. See also [22] for application of the second Lyapunov method to asymptotic analysis of equations with a small parameter.…”

Section: Change Of Variablesmentioning

confidence: 99%

“…Let us show that this solution is stable with respect to the perturbation p(z, ϕ, t). Consider ℓ(z) = z 2 /2 as a Lyapunov function candidate for equation (21). The total derivative of ℓ(z) has the form:…”

Section: Bifurcations Of the Equilibriummentioning

confidence: 99%

Stability and bifurcation phenomena in asymptotically Hamiltonian systems

Sultanov

2020

Preprint

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The influence of time-dependent perturbations on an autonomous Hamiltonian system with an equilibrium of center type is considered. It is assumed that the perturbations decay at infinity in time and vanish at the equilibrium of the unperturbed system. In this case the stability and the long-term behaviour of trajectories depend on nonlinear and non-autonomous terms of the equations. The paper investigates bifurcations associated with a change of Lyapunov stability of the equilibrium and the emergence of new attracting or repelling states in the perturbed asymptotically autonomous system. The dependence of bifurcations on the structure of decaying perturbations is discussed.

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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%

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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Bifurcations of autoresonant modes in oscillating systems with combined excitation

Cited by 10 publications

References 31 publications

Stability and bifurcation phenomena in asymptotically Hamiltonian systems

Stability and bifurcation phenomena in asymptotically Hamiltonian systems

Stability and bifurcation phenomena in asymptotically Hamiltonian systems

Damped perturbations of systems with centre-saddle bifurcation

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