ASANTE, M. Periodická okrajová úloha pro Duffingovu rovnici [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2022.

Posudek vedoucího

Šremr, Jiří

In the present thesis, the existence of a positive periodic solution to the Duffing equation is studied in both autonomous and non-autonomous cases. I would like to appreciate that the author was actively working on the thesis during the whole academic year. He composed results of consultations quite well and carried out all given tasks. On the other hand, I have, in particular, the following objections: 1. On p. 26, the fact, that the equilibrium s_1 is unstable, does not follow from Theorem 3.17. 2. On p. 26, the reason of the stability of s_2, s_3 should be explained in more details. 3. On p. 39, the proof of Theorem 4.5 is not complete; it should be explained at the very end why the last formula yields a contradiction. Conclusion: In my opinion, all the main goals of the thesis have been achieved. The text contains some misprints (commas, spaces, and capital letters) and mathematical inaccuracies, which however are mostly of minor character. The author should be more careful when presenting some information taken from the literature. Moreover, he should be more accurate in mathematical formulations. In view of the above said and the fact that the author needed supervisor's assistance in more difficult parts, I can recommend the thesis for defense with the overall classification C.

Posudek oponenta

Řehák, Pavel

The thesis deals with qualitative analysis of the Duffing-type differential equation which arises out when modeling certain oscillator. Here the oscilator contains two springs that are attached to the barriers which generally may oscillate. A phase portrait for the autonomous case is obtained. The nonautonomous case is also analyzed - in particular, the results concerning existence or uniqueness of a periodic solution are derived. The computations are supported by numerical simulations performed by MATLAB. First I mention a selection of problematic points (which however are of minor character to a large extent). The author should take care of writing commas, periods, spaces, and capital letters; there is also a certain number of other typos (e.g., p_0 in formula (4.32)). The choice of fonts is not made properly at some points. Definitions of some concepts are missing (e.g., center). Some steps in the proofs should be clarified (e.g., the line after formula (4.23) or the very end of the proof on p. 24). There are uncomplete formulas (e.g., the second line on p. 35) . Here are strengths of the thesis: I like the logical structure of the work. The most of the text is well understandable and easy to read (although it contains nontrivial considerations). Further I appreciate the combination of various sophisticated tools that are used in the proofs, such as the method of lower and upper functions, the Lyapunov inequality, and the Fredholm theorems. The numerical simulations involve carefully chosen values so that they illustrate the results quite appropriately. The author demonstrated the ability to work on a difficult topic and created an interesting text enriched with numerical simulations. The main goals have been achieved, and in view of the above said I can recommend the thesis for defense with overall classification C = good. QUESTION On page 26 it is written that the unstability of s1 follows from Theorem 3.17. Theorem 3.17 however requires positivity of the real part of all eigenvalues, and this condition is not fulfilled for s1. Please clarify.

eVSKP id 140254