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Modeling chaos 1

This compilation presents foundational texts and journals on deterministic chaos and nonlinear systems. Key works include "Deterministic Chaos: An Introduction" by H.G. Schuster and "Chaos and Fractals" by Peitgen, Jürgens, and Saupe. Topics cover one-dimensional discrete systems, the logistic equation, bifurcation diagrams, Feigenbaum constants, Lyapunov exponents, and the complexities of discrete versus continuous systems. Essential for students and researchers in physics and mathematics, this resource explores the fascinating world of chaos theory and its implications.

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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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