1. Modeling chaos 1

2. 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

3. One-dimensional discrete systems Logistic equation Mechanism of doubling the period Bifurcation diagram Doubling – period tree, Feigenbaum constants Lyapunov exponents – chaotic solutions

4. Continuous-time systems Rossler differential equation Lorenz differential equation

5. One – dimensional discrete systems

6. Bernouli function

7. Triangular function

8. Logistic function

9. Sinusoidal map

10. Iterating logistic map

11. r=2.6 x0=0.25

12. r=3.2, x0=0.25

13. x0=0.25, r=3.48

14. x0=0.2, r=4

15. Stability of equilibrium point:

16. Plot of the function: f(x)

17. f (2) ( x ) = f ( f (x) )

18. f (4) ( x ) = f ( f ( f ( f (x) ) ) )

20. Bifurcation diagram

21. r x rr Period doubling tree

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