notes on the topology of vector fields and flows
TRANSCRIPT
Notes on the Topology ofVector Fields and Flows
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8 ¢£ CU@>I TCUY@0 ^ G | e ?A@BC D i NQ@GZH&IJXTH i G a I
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→ ¡ o @ a NPGQGJ/I\IhK W @CUYRJ9C
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→ ¡ o  NPt@TÝJ
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→ ¡"[ T@ W J9e a0D NQCU@VCZY+@H;N p @ D @TCUNZJXG
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× µ → ¡ o
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φ è →
o lY+@³B TRIhNZI\CU@TBeB TRH;NQCUN T pD CUYNZI³c a B5JXTÔT a M/@IhC2J9CU@>HJ4I
φ ëì φ è ≡ φ ëZí;è pD JXGQGzIxJ9TRHC pDa Y+NPB2Y¿CUYNZINI=H;@ E T@>H ok K@C CUY@À@O/KNQtJ4GQ@TB@ H;N p @ D @T+CZNZJ9Gy@O/KRJ9CUN TI=J9TRHt@BC DE @GZH&I[;CUY@c aÔ CUY@=H;N v @ D @TCUNZJXG@O4KJXCUN TH;^&ÞîH;C = f ^ m NZIVJ9GZI B5JXGQGQ@>HÀCZY+@xâ4à 1z# Ï R 7 #93; f ^ mpoï 7 4R;35ð;#Ô#3»ÜñX 7 #93¸ &Æ41¾ .LòR 7 #93;lY@KT+NPO/K@T+@>I\I I GQKCUN TIC à okno oÄÅ ICU@GPGZInKRInN W ^ D C\J9TC8NQT pD W J9CZN TJXM KC8CUY@B T E| K D J9CZN T C D JªÇÈ@B2C D NP@>I J g© t@BC DE @GZH Ð J9BY¿^ NPT+C0^ CUY@H W JXNQTêGQNQ@>I T T@nJ9TRH TGQe T@¤C D JªÇÈ@B2C D e oq Tx^AJ D CUNQBK+GJ D [XCUYNZIlCZ@GQGZIlKRIlCUYRJ9CjC D JªÇÈ@B2C D NP@>IB;J9T=T@t@ D C D KGQezB T+t"@ D| @>[ÜB D2 I\Ij@>J9BY CZY+@ D [D M D J9TBY of a¤ C D JªÇÈ@BC D NQ@>I W J9e]4±l±F&3C B T+t"@ D| @>[ vD @J W ^&GQ@>[AN J/ICUN W @
→ ∞M CZYC D JªÇÈ@BC D NQ@>IJ9^>^ D2 JXB2YxCUY@nIJ W @¤^ NQTC oçyKCCUYNZIB;J9T TGQe BBK Da Y+@TÀCZYJXCS^ NPT+CNZINQT J9BCNQC\I a TÑC D JªÇº@BC D e JØIhC2J9CUN TJ D e^ NQTC oËmq CyNZIF@TGQN | YCU@TNQT | C IU@@ a YRJ9CyB;J9TYRJ9^&^&@Ta Y+@T a @dH D ^CUY@=B TRH;NQCZN T]CUYRJ9CCZY+@=t"@B;C DE @GZH¤M/@¤B T+CZNQTK KRIhGPexHN p @ D @T+CZNZJ9M/GP@ o TNQTCU@ D @>IhCZNQT | @uJ W ^>GQ@ fpi N | o × m B;J9TM/@wB TRIhC D KBCU@>H¤N Jt"@B2C DE @GZH¤NIFH;@ E T@>H TJXGQG ótNZJ0? f [;e m = f [ ¥ e»óUôöõ mpo5f lY@³H;@ D NPtJ9CZNQt"@∂f ¥ e&óhôöõ m Þ
∂eNI e − © ô÷õ9[ a YNQB2YÔH @>IT Cz@"NZIhC
a Y+@Te = Á [I CUYNZIyt"@B2C D¤E @GZH8NZIT CB TRCUNPT+K KRIhGQexHN p @ D @T+CZNZJ9M/GQ@ o4m
ãl@CÔJ¾M/@¯J9TeÐTK W M/@ D o lY@TÕCZY+@BK D t"@| NQt@T¨M/exC
→f C2[ f C − J m õ m NZI£J¤C D JªÇÈ@B2C D e
CUYNZIt"@B2C DLE @GZH o çK+CzIUNQT+B@n? f u[ Á m = f [ Á mpD J9GPGu["NQC*NZIBGQ@>J D CZYJXCCZY+@£pJXNZINPC\Ih@G NZIJ9GZI J¤C D JªÇº@BC D e9[XtNZJCZY+@¤BK D t"@ | NQt"@TM/eC
→f C\[ Á mpo lY+@>IU@ÓC a0 C D JªÇÈ@B2C D NP@>I§B D I\I
@>J9BY CZY+@ D o CjCZY+@^ NQTC a Y@ D @CUY@eB D II\[XC a¤ H;NZIhCZNQTB2CC D JªÇÈ@BC D NQ@>I§IhC2J D C pD2 W CZY+@¯IJ W @NQT+NPCUNZJ9GB THNQCUN TRI o lYNZIyt"N GZJ9CZ@>IyCUY@K+TNQO4K+@T@>II I GPK+CZN TRI a YNQBY W KRIhCxY GZH pD g© t"@BC DE @GZH&I oi N | K D @ × ?A@BC D i NQ@GZH0? f u[9e m = f [ ¥ e»óUôöõ m
ø ¢£ CU@>I TCUY@0 ^ G | e ?A@BC D i NQ@GZH&IJXTH i G a I
ù Ï Lâ4àÕ18 &3R 7 #93;q TCZY+@ÀB;J/Ih@ a Y+@ D @]Jt"@BC DVE @GZH T ¡xNZIH;@ E T+@>HM/e=J W J9C D NQu[ÜCUY@TxCUY@ D @NZIFJ£IhN W ^&GQ@@"^&GQNQBNQC pD W KGZJ pD CZY+@¤c a® ? oú K^&^ IU@? f ^ m = ãl^J9CA@>J9BY³^ NQTC^[ a Y@ D @ãNZIFI W @¤T × T W J9C D NQ o lY+@TxCZY+@c ab ?NZI| NQt@TwM/eJxt@ D ewIUN W ^&GP@JXTHM4@>J9KCUN KG pD W KGZJ φf ^["C m ≡ @ èvû ^ oÊ @ D @CUY@8@"^ T@T+CZNZJ9G@ü J9TdT × T W JXC D NQ³NZIyCUY@8T × T W J9C D NQH;@ E T+@>HJ4I
ý .*ª º 7 4#ª ¶1Rùj3»XñX 7 #935 &@J D @³T a NQT¶JV^ IUNQCUN T)C W J9rs@J9T)@GQ@>W @TC\J D eB2GJ/IIUN E B5JXCUN T CUY@³rsNQTRH&I C D J4Ǻ@BC D NQ@>IjCUYRJ9CjB;J9T B2BK D o @ W J9ezCUYNQTr JC D JªÇÈ@BC D e¶J4IJ W J9^&^&NPT |
→ ¡ pD WCUY@ D @>JXG]TK W M/@ D I®C Ð K+BGQNZH;@>JXT IU^J9B@ ©ªo
ù Ï ; 4Ó1¿# Ï 3»& 7 u #ª 4®#Ï 4lþ m lY@ W JX^&^&NQT | NI T@>·C £ T@ osq TCZY+NIB;J/IU@CUY@ C D JªÇÈ@B2C D e NZI J BK D t@ CZYJXCÕT@t@ DD @CUK D TRI²C a Y@ D @NPCYJ4I²M/@@T o lYNZINZI²B;J9GQGQ@>HJV3sð+43À#ª3»XñX 7 #93 å osf « D @J9M K+CCUY@>Ih@GZJ9CZ@ D oËm m lY@ W JX^&^&NQT | YRJ/IFJGQ@>J4IhCl^>@ D N Hnÿ Á [4IJXeCÌX[AJ CU@ DLa YNQBY¨NQC D @^&@>J9C\I@JXB2CZGQe a Y@ D @NPCYRJ/I£M4@@TM4@ pD @ f C Í CªÌ m = f C m pD J9GQGFC olY@C D JªÇº@BC D e W KIUC pD W JdIUN W ^&GQ@xBG IU@>HBK D t"@NQT KB2GPNZH;@>J9TáIU^J9B@ o ú K+BYáJC D JªÇÈ@B;
1. In these notes, we will often not distinguish trajectories whicharise from different initial conditions but constitute the same setof points.
eM Mk
k!k 0=
∞
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C D enNZIB5JXGQGQ@>HxJ 7 &]3¸: # o5f « D @=J9M KCCUY@>Ih@¤GZJXCU@ D o¦m
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' ¼S#ª 4 .¤ 7 ;±l#9 1ù±Fªð åú W @ K+TRH&J W @T+C2J9G+B T+B@^&C\I C ^ G | enNQT KBGQNZH;@>JXT´IU^J9B@) ¡CUYRJ9C a NQGPG0M/@KIU@>H DJ/I\IhK W @>H0Y@ D @¤NQTBGQKH@0CZY+@ p GQG a NQT | T ε ; ºð Ï :43 Ï + J¶^ NPT+C^ pD JTK W M/@ D [zH;@T CZ@>H ¢
εf ^ m [NICZY+@ÔIh@C J9GQG
^ NQTC\IJXCFJ8H;NZIhC2J9TB2@ < ε pD2 W ^ o ^ NPT+C^NILJ º º#±F # f D 7/7 °#ª `±F # m J¨IU@C`N pD @t@ D e ε > Á [¢
εf ^ m − B TC\JXNQTRI£J9CGP@>J/IhC T@^ NQTC o
T~±F;Ñ;#NIJ9TeÝIh@CVCZYJXCVNZIJ9TáJ D M/NZC D J D e8K+TN T Ih@C\I CUY@ pD W ¢
εf ^ mpo ¢£ CZ@
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¢£ CZ@>I TCUY@¤ ^ G | e ?@B2C D i NQ@GZH&IJXTH i G a I +
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εf ^ mvo lY+NZIFIUNQCUKRJ9CZN T
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! ¢£ CU@>I TCUY@0 ^ G | e ?A@BC D i NQ@GZH&IJXTH i G a I
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