The capture into parametric autoresonance

Abstract

In this work we show that the capture into parametric resonance may be explained as the pitchfork bifurcation in the primary parametric resonance equation. We prove that the solution close to the moment of the capture is described by the Painleve-2 equation. We obtain the connection formulas for the asymptotic solution of the primary parametric resonance equation before and after the capture using the matching of the asymptotic expansions.

Quantum Phenomena in a Chirped Parametric Anharmonic Oscillator

“…This nonstationary phenomenon is called parametric autoresonance (PAR). The PAR was studied in such classical oscillatory systems as the anharmonic oscillator [6,7], Faraday waves [8,9], and plasmas [10]. A related, but different control method is the direct autoresonance (AR), where instead of parametric modulations, a chirped external driving force is applied.…”

The capture of a particle into resonance at potential hole with dissipative perturbation

“…Formulas for solutions which are fit for all the regions enable one to investigate the properties of single trajectories. The papers [26,27,28,29] contain such formulas for autoresonance problems, constructed by the method of matching of asymptotic expansions [17].…”

Stability of autoresonance in dissipative systems

“…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.…”

“…x(s; ε) = ε 1/3 κ · r(τ ) cos[φ(s; α) + ψ(τ )] + O(ε 2/3 ), ε → 0, τ = ε 2/3 s, for solutions to equation (2) leads us to system (1) with the coefficients λ = 2αε −4/3 , β = δε −2/3 /2, f 1 = g 1 = −ε −2/3 A 1 /2κ, f 0 = g 0 = −A 0 /2κ, κ = 2 2/3γ. We note that in the O.A.…”

“…Submitted June 23, 2014. present problem, while studying the stability of solutions to system (1) the smallness of the coefficients of the equation is not employed. Let us clarify the features of the considered problem, in particular, the choice of a special pumping in (2). We observe that the trajectories of the unperturbed equationẍ + x − γx 3 = 0 with initial energy 0 < E < 1/(4γ) describe non-isochronous oscillations 1 .…”