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 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...
View
Full
Document
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.
 Fall '09
 BRUCH

Click to edit the document details