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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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stability of equilibrium point:
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1r=2.6
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1r=3.48
Plot of the function: f(x)
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1r=2.6
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f(2)( x ) = f ( f (x) )
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1r=3.2
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f(4)( x ) = f ( f ( f ( f (x) ) ) )
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1r=3.2
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1r=3.2
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bifurcation diagram
r
x
r
Period doubling tree
Why the discrete time logistic equation is so complicated compared to the continuous time one ?
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