Parametric resonance in coupled oscillators driven by colored noise

Abstract

The parametric resonance in two coupled oscillators driven by a Gaussian colored parametric noise is investigated. It is shown that the resonance depends essentially on the form of coupling. The phenomenon is illustrated by stability diagrams, which are obtained numerically.

Parametric resonance of two coupled spring oscillators driven by a periodic force combined with periodic noise

“…Equations ( 11) and ( 12) are Mathieu equations (for β = 0), where ω/2 = ω s + ε/2. It is known [10] that the instability regions in the parameter space (ω, σ) for the equations ( 11) and ( 12) are located close the frequencies:…”

“…Equation ( 13) represent simple resonances, and equation ( 14) represents a combination resonance [10]. In our experimental setup, a combination resonance satisfying relation ( 14) is possible.…”

“…A study [7] has shown that coupled mechanical oscillators driven with the parametric position feedback frequency close to the difference of the normal mode frequencies demonstrate instability. Parametric resonance in coupled oscillators is more dif cult to study than in single oscillators, and systematic results are limited [8][9][10]. Real excitations constantly ensure a spectrum with a nite width; hence, it is important to consider parametric noise in oscillatory systems.…”

“…Real excitations constantly ensure a spectrum with a nite width; hence, it is important to consider parametric noise in oscillatory systems. In reference [10], authors considered the noise as a parametric excitation of a system of two mutually coupled linear oscillators.…”

Stability properties of solutions of linear second order differential equations with random coefficients

“…(1.6) This is a special case of Hill's equation, which is also known as Meissner's equation [1,21,23,27,33]. Now we would like to destabilize the equilibrium position x = 0, so the problem is to find the critical values of the period 2T for which the amplitudes of all (nontrivial) motions tend to infinity as t → ∞ at arbitrarily small values of the parameter ε > 0 ("parametric resonance" [4,6] …”