Odr2 (NMAA407) -- current state of the lecture
Dalibor Pra¾ák, fall 2024
13. Dynamical systems.
Definition of dynamical system (d.s.).
Solution function as a canonical example of d.s.
Invariant set, orbit. Omega-limit set.
Alternative definition of omega-limit set: intersection
of closed forward orbits. Theorem: basic properties
of omega-limit set. Compact omega limit set as an attracting set.
Topological conjugacy. Rectification lemma.
Hyperbolic stationary points. Hartman-Grobman theorem (w/o p.)
14. La Salle's invariance principle.
Motivational example: mathematical pendulum with friction.
Review: principle of linearized (in)stability.
Lyapounov function and related stability theorems.
Orbital derivative.
p1+p2 (Oct 3) ↵
La Salle's theorem. Conclusion of the
pendulum example.
Poincaré-Bendixson theory
Remark: this theory only holds in the plane.
Definition: curve, Jordan curve, segment, transversal.
Jordan theorem (w/o p.)
Statement of the Poincaré-Bendixson theorem.
Auxiliary lemmata: existence of flowbox, monotonicity
of transversal intersections.
Transversal meets omega-limit set in at most one point.
p3 (Oct 10) ↵
Proof of the Poincaré-Bendixson theorem.
Bendixson-Dulac theorem.
p4 (Oct 17) ↵
Carathéodory theory
Review: fundamental theorem of calculus for AC and L^1 functions.
Carathéodory conditions (CAR). Carathéodory (AC) solution.
Lemma: measurability of composition of continuous and CAR function.
Lemma: integral formulation of AC solution.
Generalized Banach contraction theorem.
Generalized Picard theorem.
p5 (Oct 24) ↵
17. Sturm-Liouville theory.
18. Optimal control.
Motivational examples: moon landing problem;
weekend house problem. Abstract formulation:
controllability, time optimal problem; functional
maximization with constraints.
p6 (Oct 31) ↵
Kalman matrix. Kalman theorem about (global) controllability
of linear constant coefficient system.
Example: 1d parking problem.
Observability, duality with controllability.
Local controllability of a smooth nonlinear problem (statement only). Example: damped pendulum.
p7 (Nov 7) ↵
Proof of the local controllability theorem.
Stabilization or the problem of automatic control (``feedback'').
Lemma on spectrum for certain matrices (w/o proof).
Global stabilizability of linear problem (sketch of the proof for m=1).
Local stabilizability of nonlinear problem.
Linear problems with bounded control: problem of time optimal control.
Banach-Alaoglu theorem and its consequences.
Properties of the controllability domain.
Existence of time optimal control.
Pontryagin maximum principle: characterization of time optimal controls.
p8+9 (Nov 14) ↵
Example: exponential decay with control.
Krein-Milman theorem and its consequences. Bang-bang controls.
Sufficient conditions for global controllability (w/o proof).
p10 (Nov 21) ↵
Proof of the previous theorem.
PMP for the general control problem (with fixed time).
p11 (Nov 28) ↵
19. Bifurcation theory.
Regular point vs. bifurcation point. Examples.
Definition: bifurcation of ODEs.
Remark: non-hyperbolic stationary point - necessary
condition of bifurcation. Basic bifurcations in 1d:
saddle-node, transcritical, pitchfork.
Division lemma. Sufficient conditions for
saddle-node and transcritical bifurcation in 1d.
p12 (Dec 5) ↵
Pitchfork bifurcation in 1d (w/o proof.)
Hopf bifurcation in 2d. Remark on the normal form of Hopf bifurcation.
p13 (Dec 12) ↵
Abstract bifurcation: definition.
Theorem on bifurcation of a simple eigenvalue.
Proof of Hopf bifurcation in R^2, auxiliary lemma (without proof).
20. Invariant manifolds
Introduction to invariant manifolds.
Formulation of the preliminary problem.
Definition of invariant manifold.
Equivalence of invariance principle and reduction principle.
Lemma on bounded, unstable solutions.
Equivalence of invariance principle and fixed point principle.
Proof of the existence of invariant manifold.
Applications: localization. Centre, stable, unstable (local) manifolds.
Principle of reduced stability for centre manifold (c.m.).
Invariance of manifolds as a differential equation.
Principle of approximation for c.m. Examples.
p14 (Dec 19) ↵
Invariance of cone, stability of shadow.
Asymptotic completeness a.k.a. tracking property of c.m.
Proof of the principle of reduced stability.
p15 (Jan 9) ↵
Proof of the approximation of c.m.
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