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Unformatted text preview: 0; M pushed back to equil. • (b) : M at x = 0, has largest v > 0, overshoots • (b) to (c) M eventually comes to rest at end of SP. • (c) to (a) : M gets pulled up, eventually completes cycle. ± Diﬀerent Cases of an Uncoupled Linear System I Cosider ± ˙ x ˙ y ² = ± a1 ²± x y ² . I Solution: x ( t ) = x e at and y ( t ) = y et . • (a) a <1: x ( t ) decays faster than y ( t ); x * is stable node . • (b) a =1: symmetrical node . • (c)1 < a < 0; x ( t ) decays slower than y ( t ); stable node . • (d) a = 0: x ( t ) = x ; an line of ﬁxed points . • (e): a > 0: x ( t ) grows exponentially; x * is saddle point . xaxis, unstable manifold ; yaxis, stable manifold of x *...
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This note was uploaded on 01/04/2012 for the course CDS 280 taught by Professor Marsden,j during the Fall '08 term at Caltech.
 Fall '08
 Marsden,J

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