K 1 / p2, ú10 refers to ``Příklady a cvičení z ODR''
see here.
(Kapitola 1 / příklad 2, úloha 10 apod.)
Please keep the presentation to 10-15 minutes maximum.
It is not necessary to provide all details of elementary
calculations (limits, derivatives, computing eigenvalues etc.)
In case of any questions, do not hesitate to contact me. Short link is https://tinyurl.com/ycq32sg4 here.
============================================================ Exercise no.1 (Oct 11, 15:40 in K7) topic: dynamical systems (pcODR / Kapitola 13) K 13 / ú4 (page 3) Debora Fieberg Show that every C^1 smooth dynamical system is a solution operator to the equation of the form x'=f(x). Hint: set f(y)=dφ/dt(0,y). K 13 / ú2 (page 3), parts (i)--(iv) Lenka Hýlová Exhibit a dynamical system such that the omega-limit set of some point equals: (i) empty set, (ii) unit circle, (iii) a two-point set (iv) a line. Here ,,exhibit`` means: draw a picture of the orbit, optionally provide formula for the dynamical system and/or write the systems of ODEs which generates it. Prove Theorem 13.2. from the lecture. Filip Konopka Theorem 13.2. states that omega-limit set equals some compact set iff the orbit converges to this set in the sense of distance of sets. K 13 / ú3 (page 3), parts (i) and (ii) only Hana Marková Consider system of equations x'=-y(1-x^2) and y'=x+y(1-x^2) Find all stationary points and discuss their stability. Find the sets where x' resp. y' is positive/negative. Sketch the phase portrait of the equation. Limit yourself to |x| less than 1. ============================================================ Exercise no.2 (Oct 25, 15:40 in K7) Tomáš Bárta topic: La Salle & Poincaré-Bendixson (pcODR / Kapitola 13) K 13 / ú4 (page 8) Michelle Porth Investigate the omega-limit sets for the system x'=y-x^7(x^4+2y^2-10), y'=-x^3-3y^5(x^4+2y^2-10). Hint: apply La Salle with V=(x^4+2y^2-10)^2. K 13 / ú13 (page 11) Martin Surma Investigate the existence of periodic solutions for the system x'=ax-y+xy^2, y'=x+ay+y^3, where "a" is a real parameter. Hint: use polar coordinates. K 13 / ú15 (page 11) Radovan Švarc Let x_0 be stationary point such that the eigenvalues of the matrix linearization have positive real part. Prove that x_0 does not belong to omega-limit set of any point (except for x_0). -- Hint: apply linearized stability to system with reversed time to show that nearby points are attracted to x_0 for time going to minus infinity. ============================================================ Exercise no.3 (Nov 8, 15:40 in K7) topic: Dynamical systems & Optimal control see the problems here [pdf] ============================================================ Exercise no.4 (Nov 22, 15:40 in K7) topic: Pontryagin maximum principle K14 / příklad 3 (s. 3, zejména pak dokončení s. 24) Lenka Hýlová K14 / úloha 23 (s. 29) Hana Marková Pozn.: překlep v zadání: má být ,,odvoďte rovnici pro a(t), kde ...`` ============================================================ Exercise no.5 (Nov 29, 15:40 in K7) téma: Pontrjaginův princip maxima; bifurkace v R^1 K15 / úlohy 1(e),1(f) strana 5 Martin Surma K15 / úloha 2 strana 5 Radovan Švarc Varianta: Nechť p(x) má v bodě x_0 kořen (obecně komplexní) násobnosti k. Potom polynom q(x) s nepatrně pozměněnými koeficienty má v jistém okolí x_0 přesně k kořenů (včetně násobnosti). Návod: aplikace Rouchého věty z KA. ============================================================ Exercise no.6 (Dec 13, 15:40 in K7) Jakub Slavík téma: bifurkace v R^2, Hopfova bifurkace K15 / Příklad 7 (s. 9) Michael Zelina K15 / úloha 4c (s. 19) Mikuláš Zindulka Vyšetřete existenci periodických řešení jednak elementárně (pomocí polárních souřadnic), jednak pomocí věty o Hopfově bifurkaci resp. její normální formě. ============================================================ Exercise no.7 (Jan 3, 15:40 in K7) téma: centrální varieta K16 / Příklad 3 (s. 5) Lenka Hýlová K16 / úloha 12 (s. 11) Hana Marková ============================================================ Exercise no.8 (Jan 10, 15:40 in K7) téma: centrální varieta demo: K16 / Příklad 2 K16 / Příklad 5 (s. 7) Martin Surma K16 / úloha 13 (s. 11) Radovan Švarc