lecture2A

Lecture2A - 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

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± Two-Dimensional Linear System I Simplest class of high-dimensional systems. Important for classification of fixed points of nonlinear systems . I Two-dimensional linear system: ˙ x = ax + by ˙ y = cx + dy I Written compactly in matrix form ˙ x = A x ; A = ± a b c d ² , x = ± x y ² I Linear : if x 1 , x 2 are solutions, so is c 1 x 1 + c 2 x 2 . I x * = 0 is always a fixed point for any A . I Solutions of ˙ x = A x can be visualized as trajectories moving on ( x, y ) plane, called phase plane .
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± Vibrations of a Mass Hanging from a Linear Spring I Phase plan analysis of Simple harmonic oscillator m ¨ x + kx = 0 . I Rewritten as ˙ x = v ˙ v = - k m x = - ω 2 x I Vector Field : assign a vector ( ˙ x, ˙ v ) = ( v, - ω 2 x ) at ( x, v ).
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± Fixed Point, Closed Orbits, and Phase Portrait I Fixed point (0 , 0): mass at rest at its equilibrium position. I Closed orbit : periodic motion, oscillation of mass. (a) : x most negative, v = 0; SP most compressed. (a) to (b) : x increases, v >
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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 = ± a-1 ²± x y ² . I Solution: x ( t ) = x e at and y ( t ) = y e-t . • (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 . x-axis, unstable manifold ; y-axis, 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.

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Lecture2A - 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

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