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.

 
	
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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.


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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.

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Exercise no.3 (Nov 8, 15:40 in K7) 

topic:	Dynamical systems & Optimal control

	see the problems here [pdf]

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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 ...``

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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. 

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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.

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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 

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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