Musings on the Logistic Map Shane Celis and Yun Tao PHY 256 - - PowerPoint PPT Presentation

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Musings on the Logistic Map Shane Celis and Yun Tao PHY 256 - - PowerPoint PPT Presentation

Oct 10, 2023 43 likes •205 views

Musings on the Logistic Map Shane Celis and Yun Tao PHY 256 Hypothesis Entropy rate measures randomness Lyapunov exponent measures randomness Is there a relationship between the two? Perhaps theres a functional form something

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Musings on the Logistic Map

Shane Celis and Yun Tao PHY 256

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SLIDE 2

Hypothesis

  • Entropy rate measures randomness
  • Lyapunov exponent measures randomness
  • Is there a relationship between the two?
  • Perhaps there’s a functional form something

like this:

  • And how does partition resolution affect

the transient entropy rate?

λ = lim

p→∞ f(hµ)

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SLIDE 3

Method

  • Start simple. Use a 1D map, the beloved

logistic map.

xn+1 = r xn(1 − xn)

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Transient Uncertainty

  • f Length One

0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2 Partitions

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Transient Uncertainty

  • f Length One

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

4 Partitions

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SLIDE 6

Transient Uncertainty

  • f Length One

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 0.5 1 1.5 2 2.5 3

8 Partitions

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SLIDE 7

Transient Entropy Convergence Rate

0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Lovely

1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1 1.5 2 2.5 3 3.5 4

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100

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SLIDE 9

Entropy

2.5 2.75 3.25 3.5 3.75 4 1 2 3 4

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Entropy and Lyapunov

2.5 2.75 3.25 3.5 3.75 4 1 2 3 4 2.25 2.5 2.75 3.25 3.5 3.75 4

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100

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Bifurcation and Entropy

2.5 2.75 3.25 3.5 3.75 4 1 2 3 4 2.25 2.5 2.75 3.25 3.5 3.75 4 0.2 0.4 0.6 0.8 1

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SLIDE 12

Entropy Rate

3.2 3.4 3.6 3.8 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7

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Entropy Rate vs. Lyapunov Exponent

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100 0.1 0.2 0.3 0.4 0.5 0.6

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SLIDE 14

FOILED!

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100 0.1 0.2 0.3 0.4 0.5 0.6

really?

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SLIDE 15

Just for fun

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100 1.5 2 2.5 3 3.5 4

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SLIDE 16

Entropy vs Lyapunov Connected

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100 1.5 2 2.5 3 3.5 4