{VERSION 1 0 "X11/Motif" "1.0"}{GLOBALS 1 0}{FONT 0 "-b&h-lucidat
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{INP_R 2 0 "> "{TEXT 0 173 "with(DEtools):\012A:=[x*(1-x^2)+a*x*c
os(y),1]: a:=4;\012phaseportrait(A,[x,y],-5..5,\{[0,.5,.5],[0,-.5
,.5],[0,.8,.8],[0,.1,.1],[0,-.1,.1],[0,.1,5],[0,-.1,5]\}, title=`
limit-cycle`);"}}{OUT_R 3 0 2{DAG :3n3\`a`j2x0004}}{INP_R 4 0 "> \+
"{TEXT 0 48 "B:=[-x+a*y+x^2*y,b-a*y-x^2*y]: a:= .1 ; b:= .5 ;"}}
{OUT_R 5 0 4{DAG :3n3\`a`e3j2x0001i2x0001}}{OUT_R 6 0 4{DAG :3n3\
`b`e3j2x0005i2x0001}}{INP_R 7 0 "> "{TEXT 0 84 "phaseportrait(B,[
x,y],-5..20,\{[0,1,1],[0,1.5,.5],[0,1.2,1]\}, title=`limit-cycle \+
2`);"}}{INP_R 8 0 "> "{TEXT 0 44 "C:=[-x+a*y+x^2*y,b-a*y-x^2*y]: \+
a:= 1; b:=1 ;"}}{OUT_R 9 0 8{DAG :3n3\`a`j2x0001}}{OUT_R 10 0 8
{DAG :3n3\`b`j2x0001}}{INP_R 11 0 "> "{TEXT 0 84 "phaseportrait(C
,[x,y],-5..30,\{[0,1,1],[0,1.5,.5],[0,1.2,1]\}, title=`limit-cycl
e 2`);"}}{INP_R 12 0 "> "{TEXT 0 29 "C1:=[y,-x+c*(1-x^2)*y]; c:=1
;"}}{OUT_R 13 0 12{DAG :3n3\`C1`[2,3n3\`y`+5n3\`x`i2x0001*5+5j2x0
001pE*3p8j2x0002pApEp5pEpE}}{OUT_R 14 0 12{DAG :3n3\`c`j2x0001}}
{INP_R 15 0 "> "{TEXT 0 104 "phaseportrait(C1,[x,y],-5..10,\{[0,1
,1],[0,1.5,.5],[0,1.2,1]\}, stepsize=.2,title=`Van der Pol Unstab
le`);"}}{INP_R 16 0 "> "{TEXT 0 29 "C2:=[y,-x+c*(1-x^2)*y]; c:=3;
"}}{OUT_R 17 0 16{DAG :3n3\`C2`[2,3n3\`y`+5n3\`x`i2x0001*5+5j2x00
01pE*3p8j2x0002pApEp5pEj2x0003}}{OUT_R 18 0 16{DAG :3n3\`c`j2x000
5}}{INP_R 19 0 "> "{TEXT 0 102 "phaseportrait(C2,[x,y],-5..10,\{[
0,1,1],[0,1.5,.5],[0,1.2,1]\}, stepsize=.2,title=`Van der Pol Sta
ble`);"}}{INP_R 20 0 "> "{TEXT 0 50 "c2:=.5; c1:=1; C3:=[-c2*x+y,
(x^2/(1.+x^2))-c1*y]; "}}{OUT_R 21 0 20{DAG :3n3\`c2`e3j2x0005i2x
0001}}{OUT_R 22 0 20{DAG :3n3\`c1`j2x0001}}{OUT_R 23 0 20{DAG :3n
3\`C3`[2,3+5n3\`x`e3i2x0005i2x0001n3\`y`j2x0001+5*5p6j2x0002+5e3p
Fj2x0000pF*3p6p14pFpBpFpDpB}}{INP_R 24 0 "> "{TEXT 0 0 ""}}{OUT_R
 25 0 24{DAG :3n3\`c2`e3j2x0005i2x0001}}{OUT_R 26 0 24{DAG :3n3\`
c1`e3j2x0005i2x0001}}{INP_R 27 0 "> "{TEXT 0 136 "phaseportrait(C
3,[x,y],-3..3,\{[0,.8,.8],[0,3,3],[0,-3,-3],[0,3,-2],[0,-3,3],[0,
3,-3]\},title=`2-d Saddle Node bifurcation`, arrows=THIN);"}}
{COM_R 28 0{TEXT 1 91 "WE NEED TO KNOW THE VALUES THAT SATISFY: 2
ab=1\015SO, we can plot to see the region as before:"}}{INP_R 29 
0 "> "{TEXT 0 12 "with(plots);"}}{OUT_R 30 0 29{DAG [2,22n4\`anim
ate`n5\`animate3d`n5\`conformal`n5\`contourplot`n6\`cylinderplot`
n5\`densityplot`n4\`display`n5\`display3d`n5\`fieldplot`n5\`field
plot3d`n5\`gradplot`n5\`gradplot3d`n6\`implicitplot`n6\`implicitp
lot3d`n5\`loglogplot`n4\`logplot`n5\`matrixplot`n4\`odeplot`n5\`p
ointplot`n5\`polarplot`n5\`polygonplot`n6\`polygonplot3d`n6\`poly
hedraplot`n4\`replot`n5\`setoptions`n6\`setoptions3d`n5\`spacecur
ve`n7\`sparsematrixplot`n5\`sphereplot`n5\`surfdata`n5\`textplot`
n5\`textplot3d`n5\`tubeplot`}}{INP_R 31 0 "> "{TEXT 0 15 "f:=(1./
(2.*x));"}}{OUT_R 32 0 31{DAG :3n3\`f`+3*3n3\`x`i2x0001e3j4x00500
0000000i2x0010}}{INP_R 33 0 "> "{TEXT 0 17 "plot(f,x=.1..10);"}}
{INP_R 34 0 "> "{TEXT 0 0 ""}}}{END}