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 1.1 - Properties of omega-limit sets + Lemma 1.2
- Theorem 1.3 - Rectification theorem
- Theorem 1.4 - La Salle's invariance principle
- Theorem 1.5 - Poincaré-Bendixson theorem
- Lemma 1.6 - Flow-box theorem
- Lemma 1.7 - Intersection of an orbit and a transversal
- Lemma 1.8 - Intersection of an omega-limit set and a transversal
- Theorem 1.9 - Bendixson-Dulac criterion
2. Caratheodory theory
- Caratheodory conditions, AC solution
- Lemma 2.1 - Integrability of RHS
- Theorem 2.2 - Generalized Banach contraction theorem
- Theorem 2.3 - Generalized Picard theorem
3. Bifurcations
- point of bifurcation, saddle-node bifurcation, pitchfork bifurcation, transversal bifurctation
- Proposition 3.1 - Necessary condition for bifurcation
- Theorem 3.2 - Another necessary condition for bifurcation (idea of the proof)
- Theorem 3.3 - Sufficient condition for a saddle-node bifurcation
- Lemma 3.4 - Division lemma
- Theorem 3.5 - Sufficient condition for a transcritical bifurcation
- Theorem 3.6 - Sufficient condition for a pitchfork bifurcation (without proof)
- Theorem 3.7 - Hopf bifurcation
- Theorem 3.8 - Hopf bifurcation 2 (without proof)
4. Center manifolds
- stable, unstable, center manifold
- Theorem 4.1 - Existence of a center manifold
- Corollary 4.2 - Unstable manifold (without proof)
- Lemma 4.4 - Equivalence of (INV) and (RED)
- Lemma 4.5 - Existence of a bounded solution
- Lemma 4.6 - Equivalence of (INV) and (FP)
- Theorem 4.7 - Principle of reduced stability (without proof)
- Theorem 4.8 - Aproximation of center manifold (idea of proof)
5. Controll theory
- domain of controllability, Kalman matrix, bang-bang controll
- Theorem 5.1 - Domain of controllability of a linear system
- Theorem 5.2 - Local controllability for a non-linear problem
- Theorem 5.3 - Local controllability by bounded controlls
- Theorem 5.4 - Properties of R(t)
- Theorem 5.5 - Global controlability by bounded controlls (without proof)
- Theorem 5.6 - Existence of a bang-bang controll
- Theorem 5.7 - Existence of a time-optimal controll
- Theorem - Pontryagin maximum principle for a linear problem (without proof)
- Theorem - Pontryagin maximum principle for a non-linear problem (without proof)