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The course begins with a classification of dynamical systems based on continuity/discreteness in time and state space, and on complexity of temporal dynamics. Next, stability and bifurcation theory are introduced. In the third part, the outlined approaches are used to describe transitions of dynamics from a steady state to periodic oscillations and ultimately to chaotic oscillations, including detailed characterization of chaos using the concept of fractals and Lyapunov exponents. This methodology is applied primarily to chemical systems with complex kinetics in stirred tank reactors as well as tubular reactors. Also, connection between chaotic dynamics and hydrodynamic turbulence is discussed. The last part of the course focuses on interaction between reaction and transport, in particular on Turing and flow-distributed patterns. Applications of pattern formation to biological morphogenesis are described.
Last update: Schreiber Igor (13.11.2018)
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Student acquires an overview on methods of model analysis describing dynamical behavior of general systems with special attention to chemical systems, taking place in reactors as well as in living organisms.
An explanation is provided on properties of general nonlinear systems described by differential/difference equations, and on occurrence of nonlinear effects in chemical systems as a consequence of coupling between negative and positive feedback.
Dynamical regimes in chemical systems emerge via bifurcations(i.e., changes in the course of the system as control parameters are varying), which are systematically described.
Sustained dynamics is represented by oscillations that may be periodic, quasiperiodic or chaotic. The course also involves time series analysis with focus on measures of chaos, such as fractal dimension,Lyapunov exponens and Kolmogorov entropy.
At the end, specific chemical systems are taken to show how the outlined methods are applied. These examples include also spatially distributed systems, in particular, spontaneous pattern formation in reaction-transport systems. Last update: Schreiber Igor (13.11.2018)
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Marek M., Schreiber I.: Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge Univ. Press (1995),
Holodniok M, Klíč A., Kubíček M., Marek M.: Metody analýzy nelineárních dynamických modelů, Academia (1986),
Murray J. D., Mathematical Biology, Springer, 1989 (1st ed.), 2002 (3rd ed.)
Scott A., The Nonlinear Universe, Springer (2007). Last update: Schreiber Igor (13.11.2018)
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1. Definition of dynamical system, systems with continuous and discrete time, dissipative systems, Liouville theorem.
2. Examples of systems of chemical, hydrodynamic and biological type displaying complex dynamics.
3. Phase/state space, trajectories, asymptotic dynamics, invariant set, stability, attractors, repellors and saddles, chaotic attractor.
4. Stability of steady states, Jacobi matrix, eigenvalues/eigenvectors, stable, unstable and neutral invariant eigenspace/manifold.
5. Stability of periodic trajectories, monodromy matrix, multipliers, invariant subspaces/manifolds, homoclinic and heteroclinic orbits.
6. Structural stability. Elements of bifurcation theory, classification of bifurcations, bifurcation sequences leading to chaotic dynamics.
7. Characterization of complex dynamics (quasiperiodicity, chaos), measures of spatial and temporal complexity, fractal dimension, Ljapunov exponents, Kolmogorov entropy, classification of complex attractors.
8. Determination of measures of complex dynamics from (experimental) time series. Reconstructions of state space, data smoothing, principal component analysis. Power spectra. Specification of a student project.
9. Numerical methods for dependence of steady states or periodic orbits on a parameter - continuation and detection of bifurcations.
10. Elements of stoichiometric network analysis, identification of positive and negative feedback in complex reaction mechanisms, conditions of appearance of instabilities.
11. Applications of nonlinear analysis to (bio)chemical systems, Belousov-Zhabotinsky reaction, enzyme oscillations, biological rhythms.
12. Spatially distributed systems, complex dynamics in reaction-transport and hydrodynamic systems. Spontaneous emergence of spatial patterns, Turing bifurcation.
13. Classification of spatial and spatiotemporal patterns, applications in biology, theory of morphogenesis and differentiated growth of organisms.
14.Presentation of the solved project. Last update: Schreiber Igor (13.11.2018)
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none Last update: Schreiber Igor (13.11.2018)
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Matematics I, II and, alternatively, Methods of analysis of nonlinear dynamical models Last update: Schreiber Igor (13.11.2018)
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