Math in Moscow/2010-spring

Elementary Ordinary differential equations course by Yury Kudryashov and Ilya Schurov.

Program

1. Introduction. Examples
   1. The notion of differential equation. Solutions. Cauchy problem.
   2. Geometric approach. Phase space and extended phase space. Line field, integral curves, singular points. Simple examples.
   3. Example: fishing quota.
   4. Elementary methods of solving ODE (on exercises).
2. Existence and uniqueness
   1. Example of nonunique solution (like ẋ=x⅓)
   2. Local theorem on existence and uniqueness. (Statement.) Globalization. Continuation until the intersection with the border of the compact set in extended phase space.
3. Rectification
   1. More complicated examples and geometrical structures. Cartesian product of the systems. Multidimensional phase space.
   2. Existence and uniqueness in multidimensional case.
   3. Smooth dependence on initial condition.
   4. Coordinate change.
   5. Flow-box. Rectification. Singular points.
   6. First integrals. Full system of first integrals.
   7. Connection with linear 1st order PDE's.
4. Planar vector fields
   1. Hamiltonian systems with one degree of freedom.
   2. Perturbations, limit cycles. Hilbert's 16th problem (statement).
   3. Poincare map. Stability of limit cycles and fixed points of maps.
   4. Equation in variations.
5. Linear equations with fixed coefficients
   1. Diagonal matrix. Stable and unstable subspaces.
   2. General case. Matrix exponent. Stability.
   3. Phase portraits of linear singular points on the plane.
   4. Fundamental matrix of solutions. Liouville-Ostrogradsky Theorem.
6. Elements of dynamical system theory
   1. The connection between ODEs and dynamical systems with discrete time. (Poincare map, special flow.)
   2. Smale horseshoe. Elements of symbolic dynamics.
   3. Small oscillation. Linear ODE on the two-torus. Density of solutions. Ergodicity of irrational rotation.
7. The proof of theorem of existence and uniqueness
   1. Fixed points of contracting maps
   2. Picard approximations
   3. Smooth dependency on initial condition. Equation in variations.

Exercises

File	Date	To be returned	Deadline
Assignment 1	09/02/2010	16/02/2010	23/02/2010
Assignment 2	16/02/2010	23/02/2010	02/03/2010
Assignment 3	16/02/2010	02/02/2010	09/03/2010
Assignment 4	16/02/2010	09/02/2010	16/03/2010
Assignment 5	16/02/2010	16/02/2010	23/03/2010

Small comparison of autonomous and non-autonomous systems

	Autonomous case	Non-autonomous case
General form	ẋ=v(x)	ẋ=v(x, t)
	Phase space	Extended phase space
Geometric object	Vector field	Line field