27-dynsys - CONTINUOUS DYNAMICAL SYSTEMS I Math 21b, O....

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Unformatted text preview: CONTINUOUS DYNAMICAL SYSTEMS I Math 21b, O. Knill Homework: Section 9.1: 4,8,10,26,32,24*,46* until Tuesday CONTINUOUS DYNAMICAL SYSTMES. A differential equation d dt ~x = f ( ~x ) defines a dynamical system. The solu- tions is a curve ~x ( t ) which has the velocity vector f ( ~x ( t )) for all t . One often writes ˙ x instead of d dt x . So, we have the problem that we know a formula for the tangent at each point. The aim is to find a curve ~x ( t ) which starts at a given point ~v = ~x (0). IN ONE DIMENSION. A system ˙ x = g ( x, t ) is the general differential equation in one dimensions. Examples: • If ˙ x = g ( t ) , then x ( t ) = R t g ( t ) dt . Example: ˙ x = sin( t ) , x (0) = 0 has the solution x ( t ) = cos( t )- 1. • If ˙ x = h ( x ) , then dx/h ( x ) = dt and so t = R x dx/h ( x ) = H ( x ) so that x ( t ) = H- 1 ( t ). Example: ˙ x = 1 cos( x ) with x (0) = 0 gives dx cos( x ) = dt and after integration sin( x ) = t + C so that x ( t ) = arcsin ( t + C ). From x (0) = 0 we get C = π/ 2. • If ˙ x = g ( t ) /h ( x ) , then H ( x ) = R x h ( x ) dx = R t g ( t ) dt = G ( t ) so that x ( t ) = H- 1 ( G ( t )). Example: ˙ x = sin( t ) /x 2 , x (0) = 0 gives dxx 2 = sin( t ) dt and after integration x 3 / 3 =- cos( t ) + C so that x ( t ) = (3 C- 3 cos( t )) 1 / 3 . From x (0) = 0 we obtain C = 1. Remarks: 1) In general, we have no closed form solutions in terms of known functions. The solution x ( t ) = R t e- t 2 dt of ˙ x = e- t 2 for example can not be expressed in terms of functions exp , sin , log , √ ·...
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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