Modeling chaos 1 Books H G Schuster Deterministic

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 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

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 x 0=0. 25

r=3. 2, x 0=0. 25

x 0=0. 25, r=3. 48

x 0=0. 2, r=4

Stability of equilibrium point:

Plot of the function: f(x)

Bifurcation diagram

Period doubling tree x r r

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