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Modeling chaos 1

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 :

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Modeling chaos 1

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  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: AnInterdisciplinaryJournalofNonlinearScience, 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) ) ) )

  19. Bifurcation diagram

  20. Period doubling tree x r r

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

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