any possibly dd stfdf Eds : already it state variable. xy + 7 ulliple ordinary differential equations through form that is a first-order differential
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1 vlued Anlysis dynmicl systems : Dynmicl cn be described either systems ny ordinry differentil equtions ulliple sclr vribles in or in stte form tht is firstorder differentil dimensionl eqution vector in possibly multi stte vrible possibly spce Trnsforming ODE descriptions to stte we hve explore Eds : lredy it Suppose through done this before Let's few exmples the se description dynmicl system is given by dd stfdf y spce form : dff xy + 7 o
2 y3 % order odkfst y2dd + 30 o cn be represented s dys xy + 7 o slept : Identify the sltes + if ii y ij xy +70 X x : figfgnxitiohmgomwensnniirtiiu order 1 less thn the Mx in the A Es w : Express X s function X if ;D :p ; tken :oi Rene fn entries in X You re done! : In this dynmicl system description we i s express purely function X There " " is no u in the description input
3 y3 ' ( Two quick definitions : Order order the differentil eqution is the derivtive in tht highest differentil eqution dd st y dt ' duh + 30 / xy + 7 o order : 3 Order : 2 Order dimension dynmicl system is the the stte vrible Order : s :c hype
4 E±2 : Consider sclr dynmicl system described by it + Rx where UER is the qi If 0 liner input Express dynmicl system in the slte form i AX + Bu ie identify He sltes X nd the relevnt mtrices A X ( In :m+d this spce i Why :p ghotel nee If I B Rem h : Liner ODE 's dmit liner lwys dynmicl system description stte spce form
5 Finding system trjectories Next on our we would like to gend compute how n the initil consider system evolves over time given nd the point stte spce description input More precisely dynmicl system with the % F( n Gol : Given X( find XH This mounts to We hve two options : Anlyticlly integrting solve the kf( x u differentil if( x u eqution This in generl is unless F is difficult to do specific form Numericlly solve the eqution pproximtely
6 3 Let's consider n exmple where we cn bow solve the nlyticlly nd then differentil compre its eqution solution with numericl scheme Eye : Sclr dynmicl system x 3 strting from xco D Anlyticl solution : net I k t dt f d± / No 3 No 0 entities t RG 3 + ( ne 3 et 3 2 et
7 xlt 3 This would not be esy Rd the dynmicl ws system esinx given by sy i cot( 3 if Hence we study numericl methods Numericl solution : We will study Enter 's method to solve i * H xltl 3 line At o tht At (f z Approximte this by tttthxltt for positive At The smller the At the better the pproximtion
8 I I ( 3 Using this pproximtion xtttt xlt + It nttl Given 61 with choice let's compute ( iv At for Nt At 01 2 ( t x ( o + oil ( x ( o 3 It ( o XC 02 l ( (0^3 x l x ( I l (0^ (05 03 i I ( Notice tht ech step is pproximte nd hence error tends to ccumulte s you continue the process for lrger number itertions
9 zet o Let's solutions compre the x x 3 nlyticl nd numericl strting from ( 1 t nlyticl Xlt from Absolute error solution eg from sneer 's Euler 's method xlt 3 method Het xkyinfo 0 1 I 0 0 't O' O' Oi zz 00g bsolute error is incresing with time
10 Xlt Euler 's method for higher order systems Let X FCX U where XER " UERM Then : IR "+M " F R Replce X by Xtt + At Tt Then compute X itertively using the reltion Xttt At Xlt + At F( H Uttt Let's do n exmple suppose x n ; nd i ( III Compute At ( 1 using L ( I HxH HH4t fitment " + Criminl 't :HfH:t strting from XG 1 fittfosff :D
11 2 Another exmple : Consider the dynmicl system described by it x3 5 Compute x( 02 strting from 6t dy o using Euler 's method with step size 01 Trnsform ODE to stte spce description : XtE i (E (Ex For nottionl convenience cll z x Then X(E nd x( Exp xlod ( + Hitin oils?!%p fit + oil; Edith + to?4g totl oiled :b pl#ttoifi4i3 Red f xfo x (0^2104
12 Xe ni Equilibrium pts : Xe is n x F ( x if equilibrium pt Flxe o dynmicl system If you equilibrium pt strt the dynmicl system t n notice tht X Flxe o the remins t Xe mening system forever! Exmple : X lnd x ( ntes Eq pts stisfy ni o nd 10 Solutions re given by n 1 z ±1 io There re 2 eg pts ( I nd (4
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