p09fl42 - Lecture 42: Chaos II (11 Dec 09) lecture 41 on...

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Unformatted text preview: Lecture 42: Chaos II (11 Dec 09) lecture 41 on FPU plus the Duffing oscillator pp. 52-58 of FW supplement. A. Duffing oscillator 1. One-dimensional 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...
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p09fl42 - Lecture 42: Chaos II (11 Dec 09) lecture 41 on...

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