ARHINFUL, D. Lorenzův systém: cesta od stability k chaosu [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2020.

Posudek vedoucího

Řehák, Pavel

The thesis deals with studying various aspects (including, among others, stability and modeling)of the Lorenz system. This system has been considered in quite many works; on the other hand, there is a lack of the texts which gather information in a somehow comprehensive way and focus on relations and interpretations. I start with the selection of some problematic points. The author should take care of writing commas, periods, spaces, and capital letters. Some concepts are defined only after their first occurrence, some other ones are defined in a vague way, or are not introduced at all. In some parts, one can recognize how the text is composed from various sources without a deeper attempt on unifying a convention. One of the aims was to analyze routes to chaos - and here I feel that more a subtle investigation would be demanded. The theory of Lyapunov exponents (as well as some other interesting related topics) has not been included and applied in the thesis. Some statements without proofs (in particular, in the latter sections of Chapter 4) could have been supported at least by brief mentioning of the main ideas and/or techniques; however one needs to acknowledge that some of them are really difficult. And here is the selection of what I appreciate on the thesis. I like the readable introduction and the chapter on literature review. The exposition in some parts of Chapter 4 is given quite carefully. There is a nice summary (in the form of tables) as for stability and bifurcation with respect to the parameter r. The part concerning the description of waterwheel model is nice. English is quite good. I really appreciate the independent approach of the student in viewing the subject. The main goals have been achieved to a large extent, and so I can recommend the thesis for defense.

Posudek oponenta

Šremr, Jiří

The present thesis is focused to a study of a stability of the Lorenz system, a famous example of chaotic systems. I have, in particular, the following objections: 1. The text contains a lot of misprints and inaccuracies. 2. Chapters 3 and 4 seem to be somehow inconsistent, the terminology for the same objects differs in some parts, e.g., Definition 3.3.1. introduces the notion of equilibrium (critical) point, the terms stationary point and fixed point are used in Chapters 4.1 and 3.4, respectively. 3. Many notions are defined in a very vague sense or are not defined at all, e.g., flow on p. 22, trajectory on p. 27, orbit on p. 28, attractor on p. 29, sensitive dependence on initial conditions on p. 29. 4. On p. 32, explanation that the equilibrium (0,0,0) is the saddle point (if r>1) is not correct. 5. The title of Section 4.7 is "Chaos and Strange attractor", but strange attractor is not discussed therein at all. On the other hand, I would like to appreciate: 1. Chapter 4.5 is written very well, including illustrative examples and Table 4.1 containing summary of dependence on the parameter r. 2. Equations of motion of a waterwheel are derived correctly and precisely in Section 5.1. 3. I like how the critical point of system (5.17) are derived and interpreted in terms of dynamics of waterwheel. 4. I appreciate the description of transformation of system (5.17) to dimensionless variables which allows us to understand system (5.17) as a particular case of the Lorenz system (4.1). Conclusion: In my opinion, main goals of the thesis have been achieved. The thesis lack in some way mathematical accuracy (which should be a standard for such type of work). I would also welcome a bigger contribution of the author, as for his own observations. For example, it would be interesting to make numerical simulations of behaviour of Lorenz system when not only r, but also the other parameters are changed. In view of the above-said, I propose the evaluation D.

eVSKP id 124446