Autoresonant asymptotics in an oscillating system with weak dissipation

Abstract

We study a system of two first-order differential equations arising in averaging nonlinear systems over fast single-frequency oscillations. We consider the situation where the original system contains weak dissipative terms. We construct the asymptotic form of a two-parameter solution with an unbounded increasing amplitude. This result gives a key for understanding autoresonance in weak dissipative systems as a phenomenon of significant increase in the forced nonlinear oscillation initiated by a small external pumping.

Stability of autoresonance in dissipative systems

“…It follows from [3] that if 1 + f 1 /g 1 > 0, then there exists a two-parametric family of solutions to equations (1) with the asymptotics r(τ ) = √ λτ + O(τ 1/4 ), ψ(τ ) = o(1) as τ → ∞. Perturbed system (15) can also have autoresonance solutions with such amplitude.…”

“…We consider a mathematical model describing the initial stage of capture into autoresonance [1,2] in nonlinear oscillating systems with a small pumping [3] and dissipation: dr dτ = f (τ ) sin ψ − βr, r dψ dτ − r 2 + λτ = g(τ ) cos ψ, τ > 0, λ, β = const > 0. (1) The sought functions r(τ ) and ψ(τ ) correspond to a slowly changing amplitude and a phase shift of a fast harmonic oscillation.…”

“…2b. Such way of pumping for capturing into autoresonance of nonlinear oscillating system with dissipation was proposed in [3], where an asymptotic analysis of system (1) was made and there were obtained conditions of existence of a two-parametric family of solutions with a growing amplitude.…”

“…For model system (1) the problem of autoresonance stability is to identify a class of perturbations under which growing solution (3) is stable. In the present work we restrict ourselves by classes of perturbations under which system (4) has a global solution with initial data in the vicinity of solutions (3). Sufficient conditions ensuring this property are well-known (cf.…”

Asymptotic analysis of a model of nuclear magnetic autoresonance

“…The conditions under which autoresonance arises have been well studied for the one-dimensional nonlinear oscillators, in particular, for the Landau-Lifshits model [12]. We solve this problem here for system (3).…”