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

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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 mαq 2 + 1 4 mβq 4 Stability for q → ∞ requires β > 0 For α > 0 one stationary point dV/dq = 0, q = 0. For α < 0, 3 stationary points q = 0 ,q = ± q- α/β . This is part of the discussion of second order phase transitions, where α is taken to be a linear function of temperature α ’ a ( T- T ). 2. To see what “goes wrong” in simple perturbation treatments of the Duffing oscillator consider V ( q ) 1 2 mω 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 γ ≡ a 2 / 2 ω 2 ω τ/ 4 = Z 1 dx/ [(1- x 2 ) + γ (1- x 4 )] 1 / 2 and for small γ ω τ ’ 2 π [1- 3 4 γ ] 3. The equation of motion is ¨ q + ω 2 q + q 3 = 0 and if “linearize” naively, q = q + q 1 , the coupled equations are...
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This note was uploaded on 12/14/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at University of Wisconsin.

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

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