This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 42: Chaos II (11 Dec 09) lecture 41 on FPU plus the Duffing oscillator pp. 5258 of FW supplement. A. Duffing oscillator 1. Onedimensional nonlinear model oscillator V ( q ) = 1 2 mq 2 + 1 4 mq 4 Stability for q requires > 0 For > 0 one stationary point dV/dq = 0, q = 0. For < 0, 3 stationary points q = 0 ,q = q / . This is part of the discussion of second order phase transitions, where is taken to be a linear function of temperature a ( T T ). 2. To see what goes wrong in simple perturbation treatments of the Duffing oscillator consider V ( q ) 1 2 m 2 q 2 + 1 4 m q 4 for small The motion remains periodic (although not simply har monic) and the period for motion of amplitude a is with a 2 / 2 2 / 4 = Z 1 dx/ [(1 x 2 ) + (1 x 4 )] 1 / 2 and for small 2 [1 3 4 ] 3. The equation of motion is q + 2 q + q 3 = 0 and if linearize naively, q = q + q 1 , the coupled equations are...
View Full
Document
 Fall '09
 BRUCH

Click to edit the document details