KATEREGGA, G. Bifurcations in a chaotic dynamical system [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2019.

Posudek vedoucího

Nechvátal, Luděk

Some nonlinear dynamical systems can exhibit a complex behavior known as deterministic chaos. It means that the trajectories remain bounded in a given state space, however, they do not approach a fixed point nor a periodic orbit (or torus) but they are attracted by a complicated looking set – the so-called strange attractor. A typical feature of chaotic systems is quite prominent sensitivity to initial conditions where two trajectories starting very close to each other separate exponentially in time. This makes the system very difficult to predict from the long-term point of view. The thesis is focused on analysis (both theoretical as well as numerical) of the system introduced in the paper “Lü, Chen, Cheng, A new chaotic system and beyond: The generalized Lorenz-like system, IJBC 14 (2004), 1507–1537” with emphasis put on stability properties of the system and exploring various bifurcations. This system was chosen as it exhibits quite rich behavior (based on the choice of the parameters) and the analysis in the aforementioned paper is not quite complete. In my opinion, the topic could be handled far better, and a more thorough analysis of the system under consideration could be presented. At least, a more intense testing using the provided codes should be done. Also, I am missing a deeper theoretical insight into the problematics. The text contains some serious issues, for instance: 1) Subsection 3.6 discusses the normal form of Hopf bifurcation, however, later there is no any other conclusion on Hopf bifurcation for the investigated system; 2) Description of the algorithm for estimation of the LLE in Subsection 4.2 is misleading. The formal aspect of the thesis is also very weak. Especially, some reasonable logical structure is not provided (the particular parts do not follow each other, the text contains notions that are not further utilized and, on the other hand, some important concepts are missing or not discussed thoroughly). Besides inconsistencies (and some mistakes) in the mathematical parts, there is a huge amount of minor issues like misprints, clumsy / weird formulations, punctuation marks are ignored at many places, there are missing words / parts of the sentences, mistakes in typesetting, bad citations, the notation is not unified, etc. These flaws also do not improve the overall impression (Appendices should be a part of the thesis). Despite the above-mentioned objections, I still consider the thesis to be defendable and classify it by degree “sufficient/E”.

Posudek oponenta

Tomášek, Petr

The thesis is dealing with dynamical systems and bifurcation. Unfortunately, the thesis is found wanting for the below stated reasons: -the text is in a very poor quality and the reader has to be confused reading this text. There is a lot of grammar mistakes. The theory background is a conglomerate of several sources, which are not properly cited (there are several citations in the text, e.g. at page 26, which are not stated in the list of references), and the logical continuity vanishes in several places. Moreover, the assertions throughout the thesis are modified in such a way that the meaning is lost, e.g., p. 28. Def. 2.17, which maps trajectories of (2.1)... - there is missing a target of mapping. p. 35 Therefore when the coefficients of the characteristic polynomial P(lambda) are satisfied, ... p. 36. function V is not introduced. p. 40. The Hopf bifurcation theorem - a real chaos - assumption presented as a conclusion, the conclusion of the theorem follows just after the assertion in the plain text, in point (iii) there is introduced term alpha, which should be the term a. . Many figures are too large that they exceed the standard linewidth. Anyway, my opinion is that such a document should not figure as engineering degree thesis. The overall impression of the thesis lead me to the evaluation F/failed.

eVSKP id 117088