Modeling%20chaos%201

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Modeling chaos 1
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Books  H. G. Schuster, Deterministic chaos, an
introduction, VCH, 1995 H-O Peitgen, H.
Jurgens, D. Saupe, Chaos and fractals Springer,
1992 H-O Peitgen, H. Jurgens, D. Saupe, Fractals
for the Classroom, Part 1 and 2, Springer
1992.   Journals Chaos An Interdisciplinary
Journal of Nonlinear Science, Published by
American Institute of Physics IEEE Transactions
on Circuits and Systems, Published by IEEE
Institute
   
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One-dimensional discrete systems

• Logistic equation
• Mechanism of doubling the period
• Bifurcation diagram
• Doubling period tree, Feigenbaum constants
• Lyapunov exponents chaotic solutions

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Continuous-time systems

• Rossler differential equation
• Lorenz differential equation

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One dimensional discrete systems
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Bernouli function
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Triangular function
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Logistic function
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Sinusoidal map
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Iterating logistic map
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r2.6 x00.25
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r3.2, x00.25
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x00.25, r3.48
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x00.2, r4
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Stability of equilibrium point
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Plot of the function f(x)
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f(2)( x ) f ( f (x) )
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f(4)( x ) f ( f ( f ( f (x) ) ) )
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Bifurcation diagram
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Period doubling tree
x
r?
r
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• Why the discrete time logistic equation is so
  complicated compared to the continuous time one ?