Dynamical Systems seminar is supported by RFBR project 16-01-00748-a and Laboratory Poncelet.
Math in Moscow/2010-spring
Материал из DSWiki
- Introduction. Examples
- The notion of differential equation. Solutions. Cauchy problem.
- Geometric approach. Phase space and extended phase space. Line field, integral curves, singular points. Simple examples.
- Example: fishing quota.
- Elementary methods of solving ODE (on exercises).
- Existence and uniqueness
- Example of nonunique solution (like ẋ=x⅓)
- Local theorem on existence and uniqueness. (Statement.) Globalization. Continuation until the intersection with the border of the compact set in extended phase space.
- More complicated examples and geometrical structures. Cartesian product of the systems. Multidimensional phase space.
- Existence and uniqueness in multidimensional case.
- Smooth dependence on initial condition.
- Coordinate change.
- Flow-box. Rectification. Singular points.
- First integrals. Full system of first integrals.
- Connection with linear 1st order PDE's.
- Planar vector fields
- Hamiltonian systems with one degree of freedom.
- Perturbations, limit cycles. Hilbert's 16th problem (statement).
- Poincare map. Stability of limit cycles and fixed points of maps.
- Equation in variations.
- Linear equations with fixed coefficients
- Diagonal matrix. Stable and unstable subspaces.
- General case. Matrix exponent. Stability.
- Phase portraits of linear singular points on the plane.
- Fundamental matrix of solutions. Liouville-Ostrogradsky Theorem.
- Elements of dynamical system theory
- The connection between ODEs and dynamical systems with discrete time. (Poincare map, special flow.)
- Smale horseshoe. Elements of symbolic dynamics.
- Small oscillation. Linear ODE on the two-torus. Density of solutions. Ergodicity of irrational rotation.
- The proof of theorem of existence and uniqueness
- Fixed points of contracting maps
- Picard approximations
- Smooth dependency on initial condition. Equation in variations.
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