A pumped oscillator obeys x + 2 x + _0^2 [1+h (2 _0+ )t ]x=0

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos · Nonlinear dynamics and chaos

Modes, Packets and Media, Ch Driven and pumped nonlinear oscillators and the route to chaos

Problem

A pumped oscillator obeys

\ddot x + 2\beta\dot x + \omega_0^2\big[1+h\cos(2\omega_0+\nu)t\big]x=0,

where

h

is the fractional depth of the stiffness modulation and

\nu

is the detuning of the pump frequency from

2\omega_0

. Take

\omega_0=2000\,\mathrm{rad\,s^{-1}}

Q=\omega_0/2\beta=200

and

h=0.050

. ((a)) Using the slowly varying quadratures

x=a(t)\cos(\omega_0+\nu/2)t+b(t)\sin(\omega_0+\nu/2)t

and keeping only terms at the response frequency, derive the pair of first-order equations for

a

and

b

, and hence the growth rate of the envelope

\sqrt{a^2+b^2}

. ((b)) Derive the width, in

\nu

, of the band of pump frequencies for which the rest state is unstable (the instability tongue), and evaluate it for the numbers above.