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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart; with(DEtool
s): with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 34 " ====== \+
O is stable for r<1 ======" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "sigma:=10; b:=8/3; r:=0.9;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 196 "ODE:=\{D(x)(t)=sigma*(y(t)-x(t)),D(y)(t)=r*x(t)-y(t)-x(t)*z(t
),D(z)(t)=x(t)*y(t)-b*z(t)\};\nIC:=\{x(0)=0,y(0)=1,z(0)=20\};\nrk:=dso
lve(ODE union IC,\{x(t),y(t),z(t)\},type=numeric, method=classical[rk4
]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "odeplot(rk,[x(t),y(t
),z(t)],0..2,numpoints=200,axes=normal);" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "odeplot(rk,[[t,x(t)],[t,y(t)],[t,z(
t)/20]],0..2,numpoints=200);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {MPLTEXT 0 21 46 " ====== C+ and C- are stable for 1<r<rh ======" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "rh:=evalf(sigma*(sigma+b+3)
/(sigma-b-1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sigma:=10
; b:=8/3; r:=20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "ODE:=
\{D(x)(t)=sigma*(y(t)-x(t)),D(y)(t)=r*x(t)-y(t)-x(t)*z(t),D(z)(t)=x(t)
*y(t)-b*z(t)\};\nIC:=\{x(0)=0,y(0)=1,z(0)=20\};\nrk:=dsolve(ODE union \+
IC,\{x(t),y(t),z(t)\},type=numeric, method=classical[rk4]);" }{TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "odeplot(rk,[x(t),
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "odeplot(rk,[t,x(t)],0..80,nu
mpoints=8000,color=blue);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 29 " ====== Chaos for r>rh ======" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "sigma:=10; b:=8/3; r:=28;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 116 "ODE:=\{D(x)(t)=sigma*(y(t)-x(t)),D(y)(t)=r*
x(t)-y(t)-x(t)*z(t),D(z)(t)=x(t)*y(t)-b*z(t)\};\nIC:=\{x(0)=0,y(0)=1,z
(0)=0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "rk:=dsolve(ODE \+
union IC,\{x(t),y(t),z(t)\},type=numeric, method=classical[rk4]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "odeplot(rk,[x(t),y(t),z(t)],
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 ====== Exponential sensitivity to initial conditions ======" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "IC;\nIC1:=\{X(0)=0,Y(0)=1+1
e-5,Z(0)=0\};\nODE1:=\{D(X)(t)=sigma*(Y(t)-X(t)),D(Y)(t)=r*X(t)-Y(t)-X
(t)*Z(t),D(Z)(t)=X(t)*Y(t)-b*Z(t)\};" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 114 "rk1:=dsolve(ODE union ODE1 union IC unio
n IC1,\{x(t),y(t),z(t),X(t),Y(t),Z(t)\},type=numeric,method=classical[
rk4]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "od
eplot(rk1,[[x(t),z(t)],[X(t),Z(t)]],0..40,numpoints=4000);" }{TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "odeplot(rk1,[[t,y(t)
],[t,Y(t)]],0..40,numpoints=4000);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 98 "odeplot(rk1,[[t,log((x(t)-X(t))^2+(y(t)-Y(t)
)^2+(z(t)-Z(t))^2)]],0..40,numpoints=4000,color=blue);" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 58 " ====== Periodic window \+
in the chaotic regime r>>rh ======" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "sigma:=10; b:=8/3; r:=300;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 196 "ODE:=\{D(x)(t)=sigma*(y(t)-x(t)),D(y)(t)=r*x(t)-y
(t)-x(t)*z(t),D(z)(t)=x(t)*y(t)-b*z(t)\};\nIC:=\{x(0)=0,y(0)=1,z(0)=20
\};\nrk:=dsolve(ODE union IC,\{x(t),y(t),z(t)\},type=numeric, method=c
lassical[rk4]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 61 "odeplot(rk,[x(t),y(t),z(t)],0..5,numpoints=4000,axes=normal);
" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "odeplot(
rk,[t,y(t)],0..5,numpoints=4000,color=blue);" }{TEXT -1 0 "" }}}
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