Existence Theorems and Estimates of Solutions for Equations of Principal Resonance

Kalyakin, L. A.

doi:10.1007/s10958-014-1906-z

Cited by 7 publications

(8 citation statements)

References 6 publications

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“…The existence of solutions ρ * (τ ), ψ * (τ ) with the asymptotics (3) as τ ≥ τ * follows from [29,30]. The comparison theorems [31] applied to system (1) guarantees that the solutions can be extended to the semi-axis.…”

Section: Particular Autoresonant Solutionsmentioning

confidence: 99%

Autoresonance in oscillating systems with combined excitation and weak dissipation

Sultanov

2020

Preprint

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A mathematical model describing the initial stage of the capture into autoresonance for nonlinear oscillating systems with combined parametric and external excitation is considered. The solutions with unboundedly growing amplitude and limited phase mismatch correspond to the autoresonant capture. The paper investigates the existence, stability and bifurcations of such solutions in the presence of a weak dissipation in the system. Our technique is based on the study of particular solutions with power-law asymptotics at infinity and the construction of suitable Lyapunov functions.

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Section: Particular Autoresonant Solutionsmentioning

confidence: 99%

Autoresonance in oscillating systems with combined excitation and weak dissipation

Sultanov

2020

Preprint

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“…The existence of partial solutions as τ τ 0 , τ 0 = const > 0 with constructed asymptotics is implied by [11]. It follows from [12] that these solutions can be continued on the whole half-line τ 0. Below we discuss the stability of such solutions.…”

Section: Introductionmentioning

confidence: 95%

Stability of autoresonance in dissipative systems

Sultanov¹

2015

Ufimskii Matematicheskii Zhurnal

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Abstract. We consider a mathematical model describing the initial stage of a capture into autoresonance in nonlinear oscillating systems with a dissipation. Solutions whose amplitude increases unboundedly in time correspond to a resonance. An asymptotic expansion for such solutions is constructed as a power series with constant coefficients. The stability of autoresonance with respect to persistent perturbations is studied by means of Lapunov's second method. We describe the classes of perturbations for which a capture into autoresonance occurs.

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“…Proof. The existence of particular solutions ρ * (τ ), ψ * (τ ) with the asymptotics (3), (6), or (8) as τ ≥ τ * , τ * = const > 0 follows, for instance, from [19,20], while the comparison theorems [21] applied to system (1) guarantee that the solutions can be extended to the semi-axis τ ≥ 0.…”

Section: Particular Autoresonant Solutionsmentioning

confidence: 99%

“…Let ̺(τ ), ϕ(τ ) be a solution to system (20) starting from B ∆ 0 at τ = T 0 , where ∆ 0 = δ ε , T 0 = τ 0 (see Theorem 5). From (21) it follows that the derivative of the function v 2 (τ ) := V 2 ̺(τ ), ϕ(τ ), τ satisfies the following inequality:…”

Section: Proof Consider the Change Of Variablesmentioning

confidence: 99%

Bifurcations of autoresonant modes in oscillating systems with combined excitation

Sultanov

2019

Stud Appl Math

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A mathematical model describing the capture of nonlinear systems into the autoresonance by a combined parametric and external periodic slowly varying perturbation is considered. The autoresonance phenomenon is associated with solutions having an unboundedly growing amplitude and a limited phase mismatch. The paper investigates the behavior of such solutions when the parameters of the excitation take bifurcation values. In particular, the stability of different autoresonant modes is analyzed and the asymptotic approximations of autoresonant solutions on asymptotically long time intervals are proposed by a modified averaging method with using the constructed Lyapunov functions. K E Y W O R D Sautoresonance, asymptotics, bifurcation, nonlinear equations, stability Stud Appl Math. 2020;144:213-241.wileyonlinelibrary.com/journal/sapm

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Existence Theorems and Estimates of Solutions for Equations of Principal Resonance

Cited by 7 publications

References 6 publications

Autoresonance in oscillating systems with combined excitation and weak dissipation

Autoresonance in oscillating systems with combined excitation and weak dissipation

Stability of autoresonance in dissipative systems

Bifurcations of autoresonant modes in oscillating systems with combined excitation

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