Modeling chaos 1

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 Journals: Chaos: An Interdisciplinary Journal of Nonlinear Science, Published by American Institute of Physics IEEE 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

Bernouli function

Triangular function

Logistic function

Sinusoidal map

Iterating logistic map

r=2.6 x0=0.25

r=3.2, x0=0.25

x0=0.25, r=3.48

x0=0.2, r=4

Stability of equilibrium point:

Plot of the function: f(x)

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

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

Bifurcation diagram

r x rr Period doubling tree

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