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# homework2 - d dt ³ x y ´ = ³ 1 2 2 1 ´³ x y ´ and...

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1 CDS 140a: Homework Set 2 Due: Friday, October 16th, 2009. 1. Verify directly that the homoclinic orbits for the simple pendulum equation ¨ φ + sin φ = 0 are given by φ ( t ) = ± 2 tan - 1 (sinh t ). 2. Derive the same formula as in Exercise 1 using conservation of energy, deter- mining the value of the energy on the homoclinic orbit, solving for ˙ φ and then integrating. 3. Ignoring friction, the equations of motion for a particle in a hoop spinning about a line a distance > 0 off center are given by ¨ θ = ω 2 sin θ - R cos θ - g R sin θ . Show graphically that If ω is large enough, there will be four equilibrium points and if ω is small enough there are only two equilibrium points. 4. Find the general solution of d dt x y = 3 1 1 3 x y and sketch the phase portrait.

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Unformatted text preview: d dt ³ x y ´ = ³ 1 2 2 1 ´³ x y ´ and sketch the phase portrait. 6. Find the general solution of d dt ³ x y ´ = ³ 1 2-2 1 ´³ x y ´ and sketch the phase portrait. 7. Let T be an invertible n × n matrix and k T k be its operator norm. Show that k T-1 k ≥ 1 k T k 8. Let T be an n × n matrix satisfying k I-T k < 1. Show that T is invertible and that the series ∞ X k =0 ( I-T ) k converges to T-1 . 2 9. Write the matrix A = 2 0 0 1 2 0 0 1 2 as the sum of two commuting matrices and use this to compute e A . 10. Show that all solutions of the linear system ˙ x =-2 x-y ˙ y = x-2 y ˙ z =-z converge to the origin as t → ∞ ....
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homework2 - d dt ³ x y ´ = ³ 1 2 2 1 ´³ x y ´ and...

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