Exam requirements for ODE2 (NMMA047)
The exam consists of written (computational) and oral (theoretic) parts. To pass the exam successfully it is neccessary to gain at least 15 points (out of 30) in each of the two parts. The final grade depends on the sum of points received in the two parts
- 30 - 40 points: good
- 41 - 50 bodů: very good
- 51 - 60 bodů: excellent
Written part
The written part consists of three problems, 10 points each. The problems' topics are
- Dynamical systems
- Bifurcations
- Center manifold
- Controll theory
or their combinations. The written part takes 90 minutes and students are not allowed to use any textbooks, written notes, calculators.
Examples of the test: here.
Oral part
The oral part consists of two questions with maximum of 15 points from each of them. One question typically correspond to one subchapter from the lecture notes. The objective is to formulate basic definitions and important theorems in the field. After that, student is asked to prove a given theorem (difficulty according to the ambitions of the student) and answer supplementary questions (according to the ambitions of the student).
An example:
1. Dynamical systems in R2
Prove that any transversal intersects an omega-limit set in at most one point.
2. Carathéodory theory
Prove the generalized Banach contraction theorem.
Contents of the lectures
1. Dynamical systems
- dynamical system, orbit, omega-limit set, invariant set, topologically conjugate systems, transversal
- Theorem 2 - Properties of omega-limit sets + Lemma 1
- Theorem 3 - Distance to omega-limit set (without proof)
- Theorem 4 - Rectification lemma
- Theorem 5 - Hartman--Grobman theorem
- Theorem 6 - La Salle's principle
- Theorem 7 - Poincaré-Bendixson theorem
- Lemma 8 - Flow-box theorem
- Lemma 9 - Intersection of an orbit and a transversal
- Lemma 10 - Intersection of an omega-limit set and a transversal
- Theorem 11 - Bendixson-Dulac criterion
2. Caratheodory theory
- Caratheodory conditions, AC solution
- Lemma 14 - Integrability of RHS
- Lemma 15 - Differential and integral equations
- Theorem 16 - Generalized Banach contraction theorem
- Theorem 17 - Generalized Picard theorem
3. Bifurcations
- point of bifurcation, saddle-node bifurcation, pitchfork bifurcation, transversal bifurctation
- Lemma 18 - Division lemma
- Theorem 19 - Sufficient condition for a saddle-node bifurcation
- Theorem 20 - Sufficient condition for a transcritical bifurcation
- Theorem 21 - Sufficient condition for a pitchfork bifurcation (without proof)
- Theorem 22 - Hopf bifurcation
- Theorem 23 - Hopf bifurcation 2 (without proof)
4. Center manifolds
- stable, unstable, center manifold
- Theorem 24 - Existence of a center manifold
- Corollary 4.2 - Unstable manifold (without proof)
- Lemma 25 - Equivalence of (INV) and (RED)
- Lemma 26 - Existence of a bounded solution
- Lemma 27 - Equivalence of (INV) and (FP)
- Lemma 28 - On sets K and V (without proof)
- Theorem 29 - Tracking property (idea of proof)
- Corollary 30 - Reduced stability
- Theorem 31 - Aproximation of center manifold (without proof)
5. Controll theory
- domain of controllability, Kalman matrix, bang-bang controll
- Theorem 32 - Domain of controllability of a linear system
- Corollary 33 - Controllability of a linear system
- Theorem 34 - Local controllability for a non-linear problem
- Proposition 35 - Local controllability by bounded controlls
- Proposition 36 - Properties of R(t)
- Theorem 37 - Global controlability by bounded controlls (without proof)
- Theorem 38 - Existence of a bang-bang controll
- Theorem 39 - Existence of a time-optimal controll
- Theorem 40 - Pontryagin maximum principle for a non-linear problem (without proof)