CHAOS

Lucy Calvillo

Michael Dinse

John Donich

Elizabeth Gutierrez

Maria Uribe

Problem Statement

• Consider the function: f(x)=ax(1-x) on the interval [0,1]where a is a real number 1 < a < 5

• This function is also known as the logistic function.

Logistic Function and the unrestricted growth function

• The model for unrestricted growth is very simple: f(x) = ax

• For an example using flies this means that in each generation there will be a times as many flies as in the previous generation.

Logistic Function and the restricted growth function

• In 1845 P.F Verhulst derived a model of restricted growth.

• The model is derived by supposing the factor a decreases as the number x increases.

• The biggest population that the environment will support is x=1.

• For our example if there are x insects then 1-x is a measure of the space nature permits for population growth.

• Consequently replacing a by a(1-x) transforms the model to: f(x) = ax(1- x)

which is the initial equation we were given.

Problem Statement

• Compute the fixed points for the function: f(x)=ax(1-x)

on the interval [0,1]where a is a real

number 1 < a < 5

Fixed Points

• A fixed point is a point which does not change upon application of a map, system of differential equations, etc.

• The fixed points can be obtained graphically as the points of intersection of the curve f(x) and the line y = x.

• The fixed points of the logistic function are 0 and (a -1) / a.

Problem Statement

• Compute the first twenty values of the sequence given by:

xn+1= f(xn)Using the starting values of

x0=0.3 x0=0.6 x0=0.9

For a= 1.5, 2.1, 2.8, 3.1 & 3.6

Iterations

• Iteration: making repititions, iterations are functions that are repeated. For instance the first iteration yields:

xn+1 = f(xn) f(x) = ax (1-x) x1 = f(0.3) x1 = (1.5)(0.3)(1-0.3)x1 = 0.315

• Iterations allowed us to compare the convergence behavior.

a= 1.5 x0=0.3

0.30.3150.32366250.3283576290.3308083450.3320612760.3326948770.3330134940.333173260.3332532580.3332932860.3333133070.333323320.3333283260.333330830.3333320820.3333327070.333333020.3333331770.3333332550.333333294

10.750.50.250

0.5

0.375

0.25

0.125

0

x

y

x

y

a= 1.5 x0=0.6

0.60.360.34560.3392410.3362350.3347710.3340490.3336910.3335120.3334220.3333780.3333560.3333440.3333390.3333360.3333350.3333340.3333340.3333340.3333330.33333310.750.50.250

0.5

0.375

0.25

0.125

0

x

y

x

y

a = 1.5 x0=0.9

0.90.1350.1751630.2167210.2546290.284690.3054620.3182330.3254410.3292940.3312890.3323050.3328180.3330750.3332040.3332690.3333010.3333170.3333250.3333290.333331

10.750.50.250

0.5

0.375

0.25

0.125

0

x

y

x

y

a = 2.1 x0=0.3

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.30.4410.517690.5243430.5237560.5238150.5238090.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.52381

a = 2.1 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y0.60.5040.5249660.5236910.5238210.5238080.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.52381

a = 2.1 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.90.1890.3218860.4583780.5213620.5240420.5237860.5238120.5238090.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.52381

a = 2.8 x0=0.3

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.30.5880.6783170.6109690.6655210.6232880.657440.6305950.6522460.63510.6488950.6379250.6467350.6397130.6453450.640850.6444520.6415740.6438790.6420370.643511

a = 2.8 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y 0.60.6720.6171650.6615630.6269130.6549010.6328160.6506080.6364890.6478380.6388030.6460550.640270.6449080.6412050.6441710.6418010.6436990.6421820.6433960.642425

a = 2.8 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y0.90.2520.5277890.6978380.5904090.6771140.6121660.6647730.623980.6569610.6310170.6519370.6353630.6486950.6380910.6466060.6398180.6452620.6409170.6443990.641617

a = 3.1 x0=0.3

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y 0.30.6510.7043170.6455890.7092920.6392110.7149230.6318050.7211450.6233940.7277990.6141330.7346180.6043580.7412390.5945920.7472620.585470.7523540.5775840.75634

a = 3.1 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.60.7440.5904380.7496450.58180.7542570.5745950.757750.5690510.7602190.5650870.7618680.5624190.7629220.5607030.7635770.5596340.7639760.5589820.7642150.55859

a = 3.1 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.90.2790.6235930.7276470.6143480.7344660.604580.7410950.5948060.7471360.5856630.7522520.5777440.7562630.5714210.7591870.5667480.7611890.563520.7624920.561403

a = 3.6 x0=0.3

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y0.30.7560.664070.8030910.5692880.8827170.37270.8416610.4797630.8985260.3282380.7937920.589270.8713110.4036610.8665880.4162090.8747240.3944940.8599260.433631

a = 3.6 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.60.8640.4230140.8786640.383810.85140.4554660.892860.3443790.8128160.5477270.89180.3473750.816140.54020.8941820.3406330.8085680.5572280.888210.357455

a = 3.6 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.90.3240.7884860.6003920.8637170.4237560.8790720.3826950.8504620.4578350.8935990.3422860.8104550.5530250.8898780.3527820.8219770.5267920.8974160.3314180.797689

Problem Statement

• Compute f’(x) and explain the behavior

By evaluating the derivative at the fixed point (x*) it can be determined

•Where f ’(x*) = m, for• m < -1, the iterative path is repelled and spirals away from fixed point•-1 < m, the iterative path is attracted and spirals into the fixed point• 0 < m <1, the iterative path is attracted and staircases into the fixed point• m >1, the iterative path is repelled and staircases away

f(x) = ax(1-x)

f(x) = ax - ax2

f ’(x) = a - 2ax

f ’(x) = a (1 - 2x)

Problem Statement

• Consider g(x) = f(f(x)) and compute all fixed points.

g(x) = f(f(x))

• f(x)=ax - ax2

f(f(x))=a(ax - ax2) - a(ax - ax2)2

g(x) = f(f(x)) g(x) = a(ax - ax2) - a(ax - ax2)2

• The fixed points of the function are:0

(a - 1) / a1/2 + 1/2a + 1/2a (a2 - 2a - 3)0.5

• The first two fixed points are the same as those computed for the general logistic function.

• The two new fixed points are the numerical values of the orbit of convergence.

Problem Statement

• Investigate the sequence xn+1 = g(xn) for the values of: Using the starting values of x0=0.3

x0=0.6 x0=0.9

For a= 1.5, 2.1, 2.8, 3.1 & 3.6

a= 1.5 x0=0.3

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.30.32366250.3308083450.3326948770.333173260.3332932860.333323320.333330830.3333327070.3333331770.3333332940.3333333240.3333333310.3333333330.3333333330.3333333330.3333333330.3333333330.3333333330.3333333330.333333333

a= 1.5 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.60.34560.3362350.3340490.3335120.3333780.3333440.3333360.3333340.3333340.3333330.3333330.3333330.3333330.3333330.3333330.3333330.3333330.3333330.3333330.333333

a = 1.5 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.90.1751630.2546290.3054620.3254410.3312890.3328180.3332040.3333010.3333250.3333310.3333330.3333330.3333330.3333330.3333330.3333330.3333330.3333330.3333330.333333

a = 2.1 x0=0.3

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.30.517690.5237560.5238090.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.52381

a = 2.1 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.60.5249660.5238210.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.52381

a = 2.1 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.90.3218860.5213620.5237860.5238090.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.523810.52381

a = 2.8 x0=0.3

10.750.50.25

0.75

0.5

0.25

0

x

y

x

y

0.30.6783170.6655210.657440.6522460.6488950.6467350.6453450.6444520.6438790.6435110.6432760.6431250.6430290.6429670.6429270.6429020.6428860.6428760.6428690.642865

a = 2.8 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.60.6171650.6269130.6328160.6364890.6388030.640270.6412050.6418010.6421820.6424250.6425810.642680.6427440.6427850.6428110.6428270.6428380.6428450.6428490.642852

a = 2.8 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.90.5277890.5904090.6121660.623980.6310170.6353630.6380910.6398180.6409170.6416170.6420640.642350.6425330.642650.6427240.6427720.6428030.6428220.6428350.642843

a = 3.1 x0=0.3

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.30.7043170.7092920.7149230.7211450.7277990.7346180.7412390.7472620.7523540.756340.7592410.7612250.7625150.7633260.7638240.7641240.7643040.7644110.7644750.764512

a = 3.1 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.60.5904380.58180.5745950.5690510.5650870.5624190.5607030.5596340.5589820.558590.5583550.5582160.5581330.5580850.5580560.5580390.5580290.5580230.5580190.558017

a = 3.1 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.90.6235930.6143480.604580.5948060.5856630.5777440.5714210.5667480.563520.5614030.5600670.5592450.5587480.5584490.5582720.5581660.5581040.5580670.5580450.558033

a = 3.6 x0=0.3

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.30.664070.5692880.37270.4797630.3282380.589270.4036610.4162090.3944940.4336310.3687640.4887270.3253170.5969290.4172920.3927410.4371040.3642840.4991370.324008

a = 3.6 x0=0.6

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y0.60.4230140.383810.4554660.3443790.5477270.3473750.54020.3406330.5572280.3574550.5154050.3264580.5939340.4118510.4017460.4197430.3888460.4449770.3549610.521457

a = 3.6 x0=0.9

10.750.50.250

1

0.75

0.5

0.25

0

x

y

x

y

0.90.7884860.8637170.8790720.8504620.8935990.8104550.8898780.8219770.8974160.7976890.8763960.8564190.888170.8269660.8991750.7914730.8680840.872280.8647640.877538

Conclusions

Work Cited

http://www.ukmail.org/~oswin/logistic.html

http://www.cut-the- knot.com/blue/chaos.shtml