diplomová práce

# Periodická řešení neautonomní Duffingovy rovnice

Text práce 608.52 kB

Autor práce: Ing. Qazi Hamid Zamir

Ak. rok: 2020/2021

Vedoucí: doc. Ing. Jiří Šremr, Ph.D.

Oponent: doc. Mgr. Pavel Řehák, Ph.D.

Abstrakt:

Ordinary differential equations of various types appear in the mathematical modelling
in mechanics. Differential equations obtained are usually rather complicated nonlinear
equations. However, using suitable approximations of nonlinearities, one can derive simple
equations that are either well known or can be studied analytically. An example of such
"approximative" equation is the so-called Duffing equation. Hence, the question on the
existence of a periodic solution to the Duffing equation is closely related to the existence
of periodic vibrations of the corresponding nonlinear oscillator.

Klíčová slova:

Differential equation, Duffing equation, periodic solution, existence, uniqueness.

Termín obhajoby

30.9.2020

Výsledek obhajoby

obhájeno (práce byla úspěšně obhájena)

znakmkaBznamka

Klasifikace

B

Průběh obhajoby

Student introduced his diploma thesis to the committee members and explained the fundaments of his topic called Periodic solutions to nonautonmous Duffing equation. He answered the opponent's question satisfactorily. Question from Matteo Colangeli was about the possible extension of this topic and it was answered too.

Jazyk práce

angličtina

Fakulta

Ústav

Studijní program

Aplikované vědy v inženýrství (M2A-A)

Studijní obor

Matematické inženýrství (M-MAI)

Složení komise

prof. RNDr. Josef Šlapal, CSc. (předseda)
prof. RNDr. Miloslav Druckmüller, CSc. (místopředseda)
doc. Ing. Luděk Nechvátal, Ph.D. (člen)
doc. RNDr. Jiří Tomáš, Dr. (člen)
doc. Mgr. Pavel Řehák, Ph.D. (člen)
Prof. Bruno Rubino (člen)
Assoc. Prof. Matteo Colangeli (člen)
Assoc. Prof. Massimiliano Giuli (člen)

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. Pictures could be on a higher graphical level.
2. On p. 28, the fact, that the equilibrium (\sqrt{a/b},0) is unstable, does not follow from Theorem 3.19.
3. On p. 29, the explanation, that the equilibrium (0,0) is stable, is a little bit confusing for readers and should be refined.

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, I can recommend the thesis for defense.
Kritérium hodnocení Známka
Postup a rozsah řešení, adekvátnost použitých metod B
Vlastní přínos a originalita B
Schopnost interpretovat dosažené výsledky a vyvozovat z nich závěry C
Využitelnost výsledků v praxi nebo teorii A
Logické uspořádání práce a formální náležitosti A
Grafická, stylistická úprava a pravopis C
Práce s literaturou včetně citací A
Samostatnost studenta při zpracování tématu B

Známka navržená vedoucím: C

The thesis deals with an analysis of the Duffing differential equation which arises out when modeling nonlinear oscillators (here it is the mathematical pendulum where the pivot point is allowed to oscillate vertically). The autonomous case as well as the nonautonomous one is considered. After deriving the Duffing equation from the model, the author recalls various tools which are necessary for examining the problems concerning, in particular, periodic solutions. The core of the work is formed by Chapter 4. For example, one of the results claims, roughly speaking, that for the period T which is small enough, the Duffing equation has no non-constant T-periodic solution. The proofs utilize various sophisticated tools, such as the method of lower and upper functions, the Lyapunov inequality, and the Fredholm theorems.

I start with the selection of some problematic points (which however are mostly of minor character). The author should take care of writing commas, periods, spaces, and capital letters; there is also a certain number of typos. The author should be more careful when presenting some information taken from the sources; this concerns in particular Chapter 3. For example, Theorem 3.6 and Theorem 3.18 require f to be continuously differentiable and not just continuous. Also Theorem 3.19 needs to be fixed since in its current form it cannot be used in the computations on p. 28. The last paragraph of Section 4.1.1 is somehow unclear, and the computations should deserve a more careful treatment. Also the part of Section 4.1.2 (on p. 29) is confusing for the reader (why do we switch to such ideas without a deeper explanation?).

And here is the selection of what I appreciate in 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). The English is not bad. But what I particularly appreciate is: Some of the observations are essentialy new and the results have a good interpretation in an important model. Section 4.2 is really nice, the exposition there is given very carefully, the proofs are clever and correct, and the reader gets answers to natural questions which arise out during analyzing the problem.

The main goals have been achieved, and in view of the above said I can recommend the thesis for defense.
Kritérium hodnocení Známka