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Unformatted text preview: 1 CDS 140b: Homework Set 3 Due by Tuesday, February 10, 2009. 1. Structural stability and topological equivalence. Consider the systems ˙ x =- y ˙ y = x and ˙ x =- y + μx ˙ y = x + μy for μ 6 = 0. (a) Let F t ,G t : R 2 → R 2 denote the flows defined by these two systems. Show that for μ < lim t → + ∞ F t ( x ) 6 = 0 and lim t → + ∞ G t ( x ) = 0 for all x ∈ R 2 , x 6 = 0. (b) Show that both systems are topologically inequivalent for μ < 0. Hint: assume to the contrary that there exists a homeomorphism φ : R 2 → R 2 and a strictly nondecreasing function τ 7→ t ( τ ) from R to itself such that F t ( τ ) = φ- 1 ◦ G τ ◦ φ. Use this expression together with the limit properties derived in (a) to arrive at a contradiction. 2. A variation on a problem from the 2008 final for CDS-140a. A bead of mass m can slide along along the vertical and is attached to two walls by means of identical springs with spring constant k and natural length l . The distance between each of the walls and the vertical is given by...
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- Fall '10
- Topology, Stability theory, Bifurcation theory, Bifurcation diagram, lim Ft