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Some equilibria are stable and some are #unstable (take \+ a look at the equilibria in this example and #classify them)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "# The equilibria in any system of two equations in two" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "# unknowns can be \+ classified using the ideas we have" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "# introduced above, by using a linear approximation \+ to" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# the nonlinear syst em near an equilibrium point." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "# for example, th ere is an equilibrium at (Pi/2,0)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# near (0,Pi/2), sin(y) is approximately 1*(y-0)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "# and cos(x) is approximat ely -1*(x-Pi/2). If we do" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# a change of variables -- u = x-Pi/2, v = y," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "# we get the linearized system u' = v, v' = -u," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# which has a cente r as its critical point (which is just" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# what the picture shows)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "# we plot the linearized system." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot(\{D(x)(t) = 1*(y(t)-0),D(y)( t) = -1*(x(t)-Pi/2)\},[x(t),y(t)],t=0..1,x=-5..5,y=-5..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 353 265 265 {PLOTDATA 2 "6%-%'CURVESG6]dl7%7$$!3( Gn=\"RCkS[!#<$!3!4uV^@@%4_F*7$$!39F8)3cd$f^F*$!35fi&[yy0z%F*7$$!36&3jm ,4F9&F*$!3RL&z7A+6#\\F*7%7$$!3*H*G<.'\\fI%F*$!3VwM#RrtF?&F*7$$!3()Q(zy \")=9k%F*$!3cBl2'GEsz%F*7$$!3lr'H3.\"[>YF*$!3d.OlJN'p#\\F*7%7$$!332.r? !p1x$F*$!3+6UR!G6]>&F*7$$!3lc\\R@y1CTF*$!3**)y0'>())\\![F*7$$!3_W&Gnb2 j4%F*$!3xGV6JfgL\\F*7%7$$!3Z3uui,#[B$F*$!3)31=6()Gf=&F*7$$!36([5/5&G2O F*$!36R>))G629[F*7$$!3;*>\"o$HEJd$F*$!3_DC$*3)Q6%\\F*7%7$$!3_7(*Q(yh%) p#F*$!3][+U`XHv^F*7$$!3!\\\"3#o*=,\"4$F*$!3\\^*zlW0Z#[F*7$$!3'HS>M%f%) \\IF*$!39WAYy*y'\\\\F*7%7$$!3[Xp]p\"4<;#F*$!3W'=;&3m&G;&F*7$$!3#Q@cd$H 8vDF*$!3c8Q[\"RVr$[F*7$$!3uL-'o@Qj_#F*$!3Ap&RBMS$f\\F*7%7$$!3#Q.Ue9aZi \"F*$!3C!)>y%ff$[^F*7$$!3MdPZ!Qc%f?F*$!3w>!=_SS;&[F*7$$!3O=5g%[IC+#F*$ !3)\\;NNx>-(\\F*7%7$$!3Wj$=Vy%)y3\"F*$!3Kgl/)Gt:8&F*7$$!3ef+0jTpV:F*$! 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