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Last update: doc. Mgr. Jiří Mikšovský, Ph.D. (14.05.2023)
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Last update: doc. Mgr. Jiří Mikšovský, Ph.D. (15.02.2023)
Presentation of the basic principles of deterministic chaos & demonstration of its prominent examples. |
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Last update: RNDr. Aleš Raidl, Ph.D. (11.06.2019)
Předmět je zakončen ústní zkouškou z odpřednášené látky nebo vypracováním zkouškového projektu. |
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Last update: doc. Mgr. Jiří Mikšovský, Ph.D. (15.02.2023)
1) J. Horák, L. Krlín, A. Raidl: Deterministický chaos a jeho fyzikální aplikace, Academia, Praha, (2003), 437 str.
2) E. Ott: Chaos in dynamical systems, Cambridge University Press, Cambridge, (1993)
3) L. Smith: Chaos - A very Short Introduction, Oxford University Press, Oxford, (2007), 180 str.
4) J. Horák, L. Krlín: Deterministický chaos a matematické modely turbulence, Academia, Praha, (1996), 444 str.
5) H.D.I. Abarbanel et al.: The analysis of observed chaotic data in physical systems, Rev. Mod. Physics, 65, (1993), 1331-1392
6) Lorenz E.N.: The essence of chaos, University of Washington Press, 3. vyd. (1999)
7) J. C. Sprott: Chaos and Time-Series Analysis, Oxford University Press, Oxford, (2003) , 507 str. |
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Last update: doc. Mgr. Jiří Mikšovský, Ph.D. (15.02.2023)
Lecture |
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Last update: doc. Mgr. Jiří Mikšovský, Ph.D. (15.02.2023)
Presenting solution to exercises aimed at analysis of selected chaotic systems. |
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Last update: doc. Mgr. Jiří Mikšovský, Ph.D. (15.02.2023)
• Dynamics of one-dimensional systems: Logistic map and characteristic modes of its behavior. Cobweb plot. Bifurcation diagram. Feigenbaum constants. Stable, periodic and unstable fixed points. • Dynamics of multidimensional systems. Henon system. Transition to systems with continuous time. • Phase portrait. Poincare maps. Conservative and dissipative systems. Attracting sets, attractors and repellors. Homoclinic and heteroclinic orbits. Divergence of close trajectories in the phase space, Lyapunov exponents. • Self-similarity and fractality: Principles and examples. Mandelbrot set. Cantor set. Sierpinsky triangle/carpet. Transition from traditional to fractal dimension: Topological dimension, box-counting dimension, Hausdorff dimension, correlation dimension, Lyapunov dimension. Strange attractors. Julia & Fatou sets. • Prominent low-dimensional chaotic systems and their characteristics: Lorenz system, Rössler system. • Bifurcations: Local vs. global, subcritical vs. supercritical. Individual bifurcation types. • General definitions of chaotic behavior. • Interaction and synchronization in nonlinear and chaotic systems. • Chaos in the physics of complex systems: Examples in astrophysics, geophysics and physics of atmosphere and climate. Implications of chaos for stability and predictability. • Chaos theory in analysis of time series: Recurrence plots, phase space reconstruction, fractal dimension estimation. Estimation of the largest Lyapunov exponent.
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