1. By: Bahareh Taghizadeh
Chaos
JAN 2007
By: Bahareh Taghizadeh

2. Non-linear Dynamics
Orbit
Attractor

3. Basin of attraction
Bifurcation diagram

4. Chaos in mathematical models
Discrete
Population growth
The Logistic map
Continiouse
The Duffing oscilator
The Lorenz model

5. Population growth

6. The logistic map The effect of control parameter a= 0.9 a= 2.6 a= 3.2

7. Logistic curve :
Fixed Point

8. (Period 1)
Fixed point=0.5
Fixed point attractor
(Period 1)
(Period 1)

9. (Period 2)
(Period 2)
(Period 2)

10. (Period 4)

11. (Chaos)

12. Bifurcation Diagram for Logistic Map

13. Stable
Nuturally Stable
Unstable

14. Feigenbaum number

15. Lorenz model

18. Duffing oscilator

20. Frequency Spectrum
Structure in Phase Space
Signatures of Chaos

21. Dimensions of Dynamics
Correlation Dimension
Fractal Dimension
Information Dimension etc.
Signatures of Chaos

22. The Butterfly Effect Sensitivity to Initial Conditions
Lyapunov Exponents
Signatures of Chaos
The Butterfly Effect

23. Lyapunov Exponents
Signatures of Chaos

24. Fractal
Fractal Geometry, proposed by Benoit Mandelbrot

26. Biological systems chaos & fractal
Parkinson
Human Heart
DNA
Lung
Blood Vessels
Protein Ion Channels