Parametric autoresonance

Abstract

We investigate parametric autoresonance: a persisting phase locking which occurs when the driving frequency of a parametrically excited nonlinear oscillator slowly varies with time. In this regime, the resonant excitation is continuous and unarrested by the oscillator nonlinearity. The system has three characteristic time scales, the fastest one corresponding to the natural frequency of the oscillator. We perform averaging over the fastest time scale and analyze the reduced set of equations analytically and numerically. Analytical results are obtained by exploiting the scale separation between the two remaining time scales which enables one to use the adiabatic invariant of the perturbed nonlinear motion.

Is there a planetary influence on solar activity?

“…The strong coherence for the 208 yr band with a constant relative phase shows that solar activity and the planetary torque are phase locked. Note that phase locking arises naturally when parametric autoresonance occurs (Khain & Meerson 2001). We also observe intermittence in the signal.…”

The capture into parametric autoresonance

“…The captured asymptotic expansion is valid as ε −2/3 (1 + θ) ≫ 1. Therefore captured asymptotic solution (31) and asymptotic expansion (12) are valid both in the domain ε 2/3 ≪ (1 + θ) ≪ 1. Using the uniqueness theorem for the asymptotic expansions one can obtain that these expansions coincide in this domain.…”

“…The asymptotic behavior on the left-hand side of the validity interval In this paragraph we determine the domain of validity for asymptotic expansion (12) and match the expansion with the WKB-asymptotic expansion that was obtained in section 2. Therefore we should determine the asymptotic behavior of the Painleve-2 transcendent u(z,α,φ) as z → −∞.…”

“…Using the uniqueness theorem for the asymptotic expansions one can obtain that these expansions coincide in this domain. The usual matching procedure [20] gives us the connection formulas for the parameters A k and φ k of expansion (31) and parameters of expansion (12). For example, the parameters of the first correction term of (31) are:…”

“…Later equation (2) was studied in [11,12,13], when σ ≡ const. It was shown that the autoresonance phenomenon takes place for the primary parametric resonance equation when σ < 0 and the modulo of the coefficient |σ| grows with respect to θ.…”