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Unformatted text preview: Harmonic Oscillator (pendulum) Problem 1. In Strogatz Chapter 6 it was shown that the origin is a nonlinear
center for the pendulum example. Let x + sin x = 0. ¨ (a) Can you prove stability of the origin using linearization? Use an appropriate Liapunov function to prove that the origin is a stable xed point. (b) Lets add a damping term of the origin for x + (1 − x2 )x + sin x = 0. ¨ ˙ Study the stability > 0, < 0. Solution:
(a) x=y ˙ y = − sin x ˙
Lets look at xed point analysis not useful. Energy (0, 0). Jacobian A= 0 −1 1 0 , λ = ±i, so linear E (x, y ) = P E + KE = 1 − cos x + 1 y 2 . This is a good candidate for 2 Liapunov function, ie V (x, y ) = E (x, y ). Lets check: Let D = (−π, π ) × R. First V (0, 0) = 0 and V (x, y ) > 0 for (x, y ) ∈ D \(0, 0). ˙ V (x, y ) = = = ∂ ∂ Vx+ ˙ Vy ˙ ∂x ∂y (sin x)y − y sin x 0 a So it is (b) stable . What happens if we add a damping term to the equation? x=y ˙ y = − sin x − (1 − x2 )y ˙
Using same V we get ˙ V (x, y ) ∂ ∂ Vx+ ˙ Vy ˙ ∂x ∂y = (sin x)y + y (− sin x − (1 − x2 )y ) = − (1 − x2 )y 2 = >0
and and ˙ Take D = (−1, 1) × R, then V < 0 if D\{0, 0}. So it is assymptotically stable for >0 ˙ V >0
for if <0 for (x, y ) ∈ unstable < 0. 1 ...
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This note was uploaded on 02/28/2011 for the course MATH 104 taught by Professor Dr.buzi during the Spring '10 term at Caltech.
 Spring '10
 Dr.Buzi

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