Modeling chaos 1

Books: H. G. Schuster, Deterministic chaos, an introduction, VCH, 1995H-O Peitgen, H. Jurgens, D. Saupe, Chaos and fractalsSpringer, 1992H-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 PhysicsIEEE Transactions on Circuits and Systems, Published by IEEE Institute

  

One-dimensional discrete systems

• Logistic equation

• Mechanism of doubling the period

• Bifurcation diagram

• Doubling – period tree, Feigenbaum constants

• Lyapunov exponents – chaotic solutions

Continuous-time systems

• Rossler differential equation

• Lorenz differential equation

One – dimensional discrete systems

)x(fx n1n

]1,0[)f(x ],1,0[x nn

Bernouli function

1 mod x2)x(f

Triangular function

5.021)( xrxf

Logistic function

)x1( x r)x(f

Sinusoidal map

)xsin( r)x(f

Iterating logistic map

)x1(x rx nn1n

r=2.6 x0=0.25

r=3.2, x0=0.25

x0=0.25, r=3.48

x0=0.2, r=4

0 50 1000

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1

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1

Stability of equilibrium point:

0 0.5 10

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1r=2.6

0 0.5 10

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1r=3.48

Plot of the function: f(x)

0 0.5 10

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1r=2.6

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f(2)( x ) = f ( f (x) )

0 0.5 10

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1r=3.2

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f(4)( x ) = f ( f ( f ( f (x) ) ) )

0 0.5 10

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1r=3.2

0 0.5 10

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1r=3.2

0 0.5 10

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Bifurcation diagram

r

x

r

Period doubling tree

Why the discrete time logistic equation is so complicated compared to the continuous time one ?