vaughn climenhaga - uhclimenha/doc/pi-mu-epsilon.pdflogistic maps general systems motivating...
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![Page 1: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/1.jpg)
Logistic maps General systems
Motivating examples in dynamical systems
Vaughn Climenhaga
University of Houston
October 16, 2012
Vaughn Climenhaga (University of Houston) October 16, 2012 1 / 7
![Page 2: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/2.jpg)
Logistic maps General systems
A pretty picture
Vaughn Climenhaga (University of Houston) October 16, 2012 2 / 7
![Page 3: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/3.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 0.5
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 4: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/4.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 1
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 5: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/5.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 1.5
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 6: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/6.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 2
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 7: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/7.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 2.5
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 8: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/8.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 2.8
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 9: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/9.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 3
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 10: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/10.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 3.2
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 11: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/11.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 3.5
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 12: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/12.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 3.8
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 13: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/13.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 4
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 14: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/14.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 0.5
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 15: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/15.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 1
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 16: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/16.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 1.5
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 17: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/17.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 2
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 18: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/18.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 2.5
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 19: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/19.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 2.8
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 20: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/20.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 3
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 21: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/21.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 3.2
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 22: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/22.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 3.5
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 23: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/23.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 3.8
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 24: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/24.jpg)
Logistic maps General systems
The logistic map
Logistic map: f (x) = λx(1− x) 0 ≤ λ ≤ 4
x
f(x)
λ = 4
Maps the interval [0, 1]into itself
Iterate over and overagain: represents state ofa dynamical systemevolving in time
What is the long-term behaviour, and (how) does it depend on λ?
Vaughn Climenhaga (University of Houston) October 16, 2012 3 / 7
![Page 25: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/25.jpg)
Logistic maps General systems
More than just a pretty picture
Bifurcation diagram. Horizontal = parameter, vertical = recurrent states
Vaughn Climenhaga (University of Houston) October 16, 2012 4 / 7
![Page 26: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/26.jpg)
Logistic maps General systems
More than just a pretty picture
λ ∈ [3, 3.57 . . . ] ← period-doubling cascade
Vaughn Climenhaga (University of Houston) October 16, 2012 4 / 7
![Page 27: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/27.jpg)
Logistic maps General systems
More than just a pretty picture
λ ∈ [3.832, 3.857 . . . ] ← window of stability
Vaughn Climenhaga (University of Houston) October 16, 2012 4 / 7
![Page 28: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/28.jpg)
Logistic maps General systems
More than just a pretty picture
λ = 4 ← chaos
Vaughn Climenhaga (University of Houston) October 16, 2012 4 / 7
![Page 29: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/29.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 30: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/30.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 31: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/31.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 32: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/32.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 33: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/33.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 34: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/34.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 35: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/35.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 36: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/36.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 37: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/37.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 38: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/38.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 39: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/39.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 40: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/40.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
![Page 41: Vaughn Climenhaga - UHclimenha/doc/pi-mu-epsilon.pdfLogistic maps General systems Motivating examples in dynamical systems Vaughn Climenhaga University of Houston October 16, 2012](https://reader034.vdocuments.site/reader034/viewer/2022051923/6010edfe51d4cc3b7468c250/html5/thumbnails/41.jpg)
Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
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Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
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Logistic maps General systems
Aside: Mandelbrot set
Fix c ∈ C: let z0 = 0, zn+1 = z2n + c .
Mandelbrot set M = {c | zn 6→ ∞}
Vaughn Climenhaga (University of Houston) October 16, 2012 5 / 7
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Logistic maps General systems
General questions
Numerical picture of bifurcation diagram for logistic maps raises questions:
1 Various phenomena are suggested by numerics: period-doublingcascades, windows of stability, self-similarity, chaos. Can theirexistence be proved rigorously?
2 Qualitative behaviour depends on parameter. How large are theparameter sets on which different behaviours occur?
3 Can consider other one-parameter families of interval mapsfλ : [0, 1] . Does the same story happen here?
4 What about higher dimensions (fλ : Rd ) or manifolds (fλ : M )?
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Logistic maps General systems
General answers
1 Rigorous phenomena for logistic maps.Period-doubling, windows of stability, self-similarity, chaos: All canbe rigorously proved to exist.
2 Size of parameter sets (prevalence of different behaviours).
Windows of stability are open and dense. Chaos is a Cantor set buthas positive measure.
3 Other interval maps.
Same results can be proved for very general families of interval maps.Even some quantitative results are identical, such as rate of conver-gence in period-doubling cascades (Feigenbaum universality).
4 Higher dimensions.
Numerics suggest a similar story, but proofs are much harderand most answers are still unknown.
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Logistic maps General systems
General answers
1 Rigorous phenomena for logistic maps.Period-doubling, windows of stability, self-similarity, chaos: All canbe rigorously proved to exist.
2 Size of parameter sets (prevalence of different behaviours).Windows of stability are open and dense. Chaos is a Cantor set buthas positive measure.
3 Other interval maps.
Same results can be proved for very general families of interval maps.Even some quantitative results are identical, such as rate of conver-gence in period-doubling cascades (Feigenbaum universality).
4 Higher dimensions.
Numerics suggest a similar story, but proofs are much harderand most answers are still unknown.
Vaughn Climenhaga (University of Houston) October 16, 2012 7 / 7
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Logistic maps General systems
General answers
1 Rigorous phenomena for logistic maps.Period-doubling, windows of stability, self-similarity, chaos: All canbe rigorously proved to exist.
2 Size of parameter sets (prevalence of different behaviours).Windows of stability are open and dense. Chaos is a Cantor set buthas positive measure.
3 Other interval maps.Same results can be proved for very general families of interval maps.Even some quantitative results are identical, such as rate of conver-gence in period-doubling cascades (Feigenbaum universality).
4 Higher dimensions.
Numerics suggest a similar story, but proofs are much harderand most answers are still unknown.
Vaughn Climenhaga (University of Houston) October 16, 2012 7 / 7
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Logistic maps General systems
General answers
1 Rigorous phenomena for logistic maps.Period-doubling, windows of stability, self-similarity, chaos: All canbe rigorously proved to exist.
2 Size of parameter sets (prevalence of different behaviours).Windows of stability are open and dense. Chaos is a Cantor set buthas positive measure.
3 Other interval maps.Same results can be proved for very general families of interval maps.Even some quantitative results are identical, such as rate of conver-gence in period-doubling cascades (Feigenbaum universality).
4 Higher dimensions.Numerics suggest a similar story, but proofs are much harderand most answers are still unknown.
Vaughn Climenhaga (University of Houston) October 16, 2012 7 / 7