KHAN, R. Zobecněná logistická zobrazení [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2025.

Posudek vedoucího

Řehák, Pavel

The thesis analyzes Richard's discrete logistic equation, which, unlike its continuous counterpart, exhibits highly complex dynamics. Specifically, the study of the modified logistic map covers the following topics: stability of equilibria, periodic cycles and period-doubling, bifurcation, chaotic behavior, and Lyapunov exponents. The text is logically structured. The theoretical framework is properly introduced. The work includes numerous numerical simulations. Several working drafts were written before the final version was completed. Unfortunately, the final version is not entirely satisfactory. Below is a selection of some shortcomings. The author should be more careful in distinguishing whether a value is exact or approximate and what conclusions can be drawn from numerical simulations. For example, on page 25, the value 0.5773 is referred to as a fixed point, yet it is an approximate value, while the exact value can be calculated using the formula on page 22. The author could have attempted more iterations in Table 3, which relates to Feigenbaum's number, as it is defined as a limit value. The section on Lyapunov exponents could have been more thoroughly developed. For instance, a zoomed-in image or a more detailed description would have been beneficial. It is unclear, for example, to which initial value the graphical simulation on page 48 corresponds. Additionally, it is unknown where the value 0.5 in bottom of the page 47 comes from and why it is critical. It should be noted that the text partially overlaps with a diploma thesis (reference [5]), which investigated the classical case of the logistic map. This is understandable, as the same theoretical framework was used and needed to be introduced. However, the author could have explored other literature more extensively and been more innovative in writing these sections. Unfortunately, the author was overly influenced by the referenced work in other parts as well, such as the commentary in the section on Lyapunov exponents. On the positive side, I would highlight the choice of the topic itself and the bold attempt to address it. Unlike the classical case, which is covered in many texts, very few works can be found on this topic. Moreover, Richard's logistic map is substantially more computationally demanding. The author also handled the numerical simulations reasonably well (aside from some interpretations). The author demonstrated some ability to work on an advanced topic and produced, with several exceptions, a meaningful text that provides both qualitative and quantitative analysis of the modified logistic map. The main objectives have been achieved, and in view of the above, I recommend the thesis for defense with an overall grade of D.

Posudek oponenta

Opluštil, Zdeněk

The diploma thesis studies the topic of generalized logistic maps, focusing particularly on Richard’s logistic map and its dynamical properties. The student provides an introduction to both differential and difference equations, analyzes stationary points for both the classical logistic and Richard’s equations, and presents numerical simulations. While the author demonstrates effort in the numerical simulation and graphical representation of the considered logistic maps, the thesis contains mathematical inaccuracies and misunderstandings. I mention some of them below - Throughout the thesis, the author works with model parameters without carefully specifying the conditions under which the models are valid. For example, on page 15, the derivation of solution x(t) assumes that the carrying capacity K is greater than the population size x, but this restriction is not mentioned, and the derivation is not correct. - On page 23, the author finds a stationary point for the growth parameter “mu” but does not verify whether this point is a maximum. - On pages 25 and 26, in Tables 1a and 1b, the author rounds the computed iteration values of y to four decimal places. Based on these approximated values, the author incorrectly claims that sequences converging to, for example, y=0.5773 represent equilibrium points. This conclusion is flawed because the rounding iterations do not guarantee convergence to a fixed point. - On page 29, the author analyzes the stability of equilibria for only three specific values of the growth parameter “mu”. However, it is possible to derive the character of the equilibria for all values of “mu” greater than 1, except for critical points, based on Lemma 2.10. - On page 38, the sets of “basins of attraction” are determined by rounded iteration values. This is not correct because the rounding changes the “behavior” of iterations. Moreover, the constructed sets do not match the results presented in Table 2. - On page 47 below, the "Remark" presented does not make any sense in the given context. It appears disconnected from the surrounding discussion.  The greatest contribution of the diploma thesis lies in the numerical simulations and graphical analyses, which effectively illustrate the behavior of solutions to the considered difference equations. The author demonstrated good command of numerical solvers and successfully applied them to simulate and visualize the dynamics of generalized logistic maps, a topic that is not trivial to study. However, the most interesting and advanced parts of the topic (namely, Chapters 3 and 4, which focus on period-doubling, bifurcations, and chaotic behavior) are unfortunately given too little attention. These chapters should have been more developed, both numerically and theoretically. The main objectives of the diploma thesis were fulfilled. Despite the shortcomings mentioned earlier, I recommend the thesis for defense and evaluate it with a grade of D.

eVSKP id 165711