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Chaos in continuous-time and discrete-time systems. Bifurcation diagrams, local and global bifurcations. Fractals and fractal dimension, (strange) attractors, Lyapunov exponents. Chaos control. Manifestations of chaotic behavior in the physics of complex systems and in climate theory.
Last update: Mikšovský Jiří, doc. Mgr., Ph.D. (14.05.2023)
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Presentation of basic principles of deterministic chaos & demonstration of its prominent examples. Last update: Mikšovský Jiří, doc. Mgr., Ph.D. (14.01.2025)
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Presentation of an assigned course-specific project by the student, followed by an oral exam,. Last update: Mikšovský Jiří, doc. Mgr., Ph.D. (14.01.2025)
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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. Last update: Mikšovský Jiří, doc. Mgr., Ph.D. (15.02.2023)
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Lecture Last update: Mikšovský Jiří, doc. Mgr., Ph.D. (15.02.2023)
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Presenting solution to exercises aimed at analysis of selected chaotic systems. Last update: Mikšovský Jiří, doc. Mgr., Ph.D. (15.02.2023)
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• 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.
Last update: Mikšovský Jiří, doc. Mgr., Ph.D. (15.02.2023)
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