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Fisher-Price

# Fisher-Price - c 2 e rt 1 c 2 e rt Finally dividing top and...

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The Logistic Flow (continuous) Tom Carter Complex Systems Summer School June, 2009 1

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Discrete logistic map We all know that the discrete logistic map P n + 1 = rP n ( 1 - P n ) exhibits interesting behavior of various sorts for various values of the parameter r , including chaos, etc. 2
Discrete logistic map – bifurcation diagram 3

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Continuous logistic flow What kind of behavior can we expect from a continuous version of a logistic flow: dP dt = rP ( 1 - P ) ? 4
Continuous logistic flow - solving Note that this is a non-linear ODE, but fortunately we can actually integrate . . . dP dt = rP ( 1 - P ) dP P ( 1 - P ) = rdt Thus: Z dP P ( 1 - P ) = Z rdt Z dP P ( 1 - P ) = rt + c 1 5

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Continuous logistic flow - solving By partial fractions, we have: Z dP P + Z dP ( 1 - P ) = rt + c 1 log ( P ) - log ( 1 - P ) = rt + c 1 log ( P 1 - P ) = rt + c 1 P 1 - P = e rt + c 1 P 1 - P = c 2 e rt 6
Continuous logistic flow - solving This gives us: P = ( 1 - P ) c 2 e rt And thus: P = c 2 e rt - Pc 2 e rt P + Pc 2 e rt = c 2 e rt P ( 1 + c 2 e rt ) = c 2 e rt 7

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Continuous logistic flow - solving From this we get:

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Unformatted text preview: c 2 e rt 1 + c 2 e rt Finally, dividing top and bottom by c 2 e rt and simplifying, we have: P = 1 1 + ce-rt 8 The classic logistic/sigmoid curve and changes in c and r make minor changes in the behavior near 0 . . . 9 Discrete vs. Continuous The difference between the behavior of the discrete and continuous logistic functions can give us some idea of the signiﬁcance of working in the discrete regime . . . 10 xy-pic test page A = ± / B ± B / C λω λ C λ 2 λ P 2 λω λ P ω λ → λ P • 1 ² 2 ! 3 ’ x x x x x x 11 Fin . . . Slides for this talk will be available at: http://csustan.csustan.edu/~tom/SFI-CSSS/2009 The Logistic Flow (continuous) Tom Carter Complex Systems Summer School June, 2009 12...
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