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

This preview shows pages 1–4. Sign up to view the full content.

± 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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
± 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 ).
± 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 >

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 *...
View Full Document

## 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.

### Page1 / 4

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online