Math 2280-1Tuesday April 12Flavor of section 6.5: CHAOStransition to chaos in the discrete logistic equation, pages 430-434, involves "period doubling", which has some universal characteristics.restart:f:=(x,r)->r*x*(1-x);
for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIjRGJ0YvRi8= this function maps the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjRGNEY0LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRjQ= to a subinterval of 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For any such fixed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5we use f to define a nonlinear discrete dynamical system on this interval:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1GOzYkUSQwLjVGJ0Y+Rj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRIm5GJ0YyRjUtSSNtb0dGJDYtUSIrRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIxRidGQUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GPjYtUSI9RidGQUZDRkZGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRlxvLUYvNiVRImZGJ0YyRjUtSShtZmVuY2VkR0YkNiQtRiM2Ji1GLzYlUSJyRidGMkY1LUY+Ni1RIixGJ0ZBRkMvRkdGNEZIRkpGTEZORlAvRlNRJjAuMGVtRicvRlZRLDAuMzMzMzMzM2VtRictRiw2JUYuLUYjNiVGOkYyRjVGZW5GQUZBRkE=we're interested in whether this dynamical system creates periodic orbits in the limit as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYuUSgmaW5maW47RidGL0YyRjtGPkZARkJGREZGRkhGSkZNLUY2Ni5RIi5GJ0YvRjJGO0Y+RkBGQkZERkZGSEZKRk1GOQ== Notice there is always one fixed point, namely the solution to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZzZG90O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInhGJ0YvRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZpbkZPRjlGOS1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORl9vRk9GOQ==which is 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However, this fixed point may not be stable.with(plots):r0:=0.0;for i from 0 to 400 do
c:=r0+.01*i: #current r-value being tested, ranges from
#0.0 to 4.0 in this case
t:=0.5: #initial value for the discrete dynamical system
for j from 1 to 1000 do
t:=f(t,c):
end do: #use the first 1000 steps to let the system
#converge to a steady periodic solution, if
#it's going to.
for j from 1 to 100 do
t:=f(t,c):
P[100*i+j]:=[c,t]: #then record the next 100 values
od:
od:
pointplot({seq(P[k],k=1..40000)},symbol=point); #discuss with cobwebbing.restart:f:=(x,r)->r*x*(1-x);with(plots):r0:=3.0; #focus on r between 3 and 4for i from 0 to 400 do
c:=r0+.0025*i:
t:=0.5:
for j from 1 to 1000 do
t:=f(t,c):
end do:
for j from 1 to 100 do
t:=f(t,c):
P[100*i+j]:=[c,t]:
od:
od:
pointplot({seq(P[k],k=1..40000)},symbol=point);restart:f:=(x,r)->r*x*(1-x);with(plots):r0:=3.4;for i from 0 to 400 do
c:=r0+.00025*i:
t:=0.5:
for j from 1 to 1000 do
t:=f(t,c):
end do:
for j from 1 to 100 do
t:=f(t,c):
P[100*i+j]:=[c,t]:
od:
od:
pointplot({seq(P[k],k=1..40000)},symbol=point);JSFHrestart:f:=(x,r)->r*x*(1-x);with(plots):r0:=3.5;for i from 0 to 400 do
c:=r0+.00025*i:
t:=0.5:
for j from 1 to 1000 do
t:=f(t,c):
end do:
for j from 1 to 100 do
t:=f(t,c):
P[100*i+j]:=[c,t]:
od:
od:
pointplot({seq(P[k],k=1..40000)},symbol=point);QyYqJi1JIitHJSpwcm90ZWN0ZWRHNiQkIiVbTSEiJEYqIiIiLUYlNiQkIiVWTkYqJCEkWCQhIiMhIiJGKyomRixGKy1GJTYkJCIla05GKiQhJVZORipGM0Yrrestart:f:=(x,r)->r*x*(1-x);with(plots):r0:=3.8;for i from 0 to 400 do
c:=r0+.00025*i:
t:=0.5:
for j from 1 to 1000 do
t:=f(t,c):
end do:
for j from 1 to 100 do
t:=f(t,c):
P[100*i+j]:=[c,t]:
od:
od:
pointplot({seq(P[k],k=1..40000)},symbol=point);
#period 3, doubling to 6, 12, ... chaosrestart:f:=(x,r)->r*x*(1-x);with(plots):r0:=3.7;for i from 0 to 400 do
c:=r0+.00025*i:
t:=0.5:
for j from 1 to 1000 do
t:=f(t,c):
end do:
for j from 1 to 100 do
t:=f(t,c):
P[100*i+j]:=[c,t]:
od:
od:
pointplot({seq(P[k],k=1..40000)},symbol=point);
#period 5, doubling to 10, ... chaosJSFHrestart:f:=(x,r)->r*x*(1-x);with(plots):r0:=3.77;for i from 0 to 400 do
c:=r0+.000025*i:
t:=0.5:
for j from 1 to 1000 do
t:=f(t,c):
end do:
for j from 1 to 100 do
t:=f(t,c):
P[100*i+j]:=[c,t]:
od:
od:
pointplot({seq(P[k],k=1..40000)},symbol=point);
#period 7, doubling to 14, ... chaosJSFH