Syllabus
1. Dynamical system. Orbit, stationary point, invariant set. Alpha- and omega-limit sets and their properties. La Salle invariance principle. Conjugate dynamical systems. Lemma on rectifications. Poincaré-Bendixson theory in the plane. Bendixson-Dulac criterion of non-existence of periodic solutions.
2. Carathéodory theory - notion of an absolutely continuous solution, local existence and uniqueness.
3. Optimal control theory.
4. Bifurcations. Basic types of bifurcations. Sufficient conditions for existence of bifurcations. Hopf bifurcation.
5. Stable, unstable and central manifolds. Invariance principle and its reformulations. Existence of the central manifold. Approximation of the central manifold. Reduced stability principle. Hartman-Grobman theorem.
Annotation
Mandatory course for the master study branch Mathematical analysis. Recommended for the first year of master studies. Devoted to advanced topics in theory of ordinary differential equations. Content: dynamical systems;
Poincaré-Bendixson theory; Carathéodory theory; optimal control, Pontryagin maximum principle; bifurcation; stable, unstable and central manifolds.