The document presents various simulations and analyses related to logistic equations and chaotic systems, highlighting time evolution, return maps, and the significance of initial conditions. It compares findings with Monte Carlo simulations and discusses determinants of chaos. It also lists multiple sources for further reading on the subject matter.

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3. My MATLAB Simulation Data
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4. My MATLAB Simulation Data
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5. Time evolution of the overlaps:
3 0 0 1 0
3 0 0 1 0
3 0 0 1 0
3 0 0 1 0
3 0 0 1 0
3 0 0 1 0
0 3 1 0 1
0 3 1 0 1
0 3 1 0 1
0 3 1 0 1
0 3 1 0 1
0 3 1 0 1
0 1 3 0 0
0 1 3 0 0
0 1 3 0 0
0 1 3 0 0
0 1 3 0 0
0 1 3 0 0
1 0 0 3 0
1 0 0 3 0
1 0 0 3 0
1 0 0 3 0
1 0 0 3 0
1 0 0 3 0
0 1 0 0 3
0 1 0 0 3
0 1 0 0 3
0 1 0 0 3
0 1 0 0 3
0 1 0 0 3
3 0 0 1 0
3 0 0 1 0
3 0 0 1 0
3 0 0 1 0
3 0 0 1 0
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6. Analyzing the Distributions
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7. Analyzing the Distributions
Compare with a Monte Carlo simulation:
But of course, the sequence also matters.
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8. Logistic Equation: Time Series Plot
xi+1 = r xi (1-xi)
f(n)
n
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9. Logistic Equation: Graphical Iteration
Pts. on the return map
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10. Logistic Equation: First Return Map
f(n+1)
f(n)
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11. Logistic Equation: Return Map 2
f(n+2)
f(n)
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12. Logistic Map: Return Map 3
f(n+3)
f(n)
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13. Logistic Equation: Return Map 4
f(n+4)
f(n)
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14. Logistic Equation: Return Map 5
f(n+5)
f(n)
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15. Neural Network: First Return Map
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16. Determinants of Chaos
• Autocorrelation function
• Return map
• Sensitive dependence on initial conditions
• Unstable Periodic Orbits
• Response to Chaos Control and Anticontrol
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17. Sources
Physical Review E
pre.aps.org
Nature
nature.com
Chaos: the making of a new science
James Gleick
Python programming language
python.org
MATLAB® computing language
mathworks.in/products/matlab
Univ. of Yale online resources on chaos
classes.yale.edu/fractals/chaos/welcome.html
California State Univ. East Bay Hayward Statistics Dept. online resources
sci.csueastbay.edu/statistics/Resources/Essays/PoisExp.htm
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