Differential Equations and Stability Theory
This is an introduction to the qualitative theory of differential equations. The subject gives a mathematical background to the nonlinear control. This subject is also an introduction to partial differential equations of the second order.
- Define fundamental notions related to qualitative theory of ordinary differential equations and fundamental notions related to partial differential equations of second order.
- Relate knowledge achived in linear algebra to behaviour of linear and nonlinear systems
- Compare knowledge from other courses with mathematical knowledge about differential equations.
- Recognize difference between analytic, numeric and qualitative approach to differential equations.
- Analyze a problem, he/she will be able to modify the methods and apply to them
- Estimate is his/her solution in accordance to theoretical basis
Forms of Teaching
Midexam, exam, seminarConsultations
|Type||Threshold||Percent of Grade||Comment:||Percent of Grade|
|Homeworks||0 %||10 %||0 %||0 %|
|Class participation||0 %||4 %||0 %||0 %|
|Seminar/Project||0 %||6 %||0 %||0 %|
|Mid Term Exam: Written||0 %||35 %||0 %|
|Final Exam: Written||0 %||45 %|
|Exam: Written||0 %||100 %|
Week by Week Schedule
- Eigenvalues and the Jordan form of a matrix. Linear systems of differential equations in the plane.
- Klasifikacija faznih portreta linearnih sustava oko Classification of phase portraits of linear systems near the equilibrium points. Stability and asymptotical stability.
- Closed trajectory and limit cycle. Linearization, Hartman-Grobman theorem.
- Stability theory. Invariant manifolds.
- Bendixson theorem, Duffing equation. Poincare index theory
- Dissipative and Conservative systems, energy method. Liouville theorem.
- Lyapunov stability and instability theorems. Poincare Bendixson theorem.
- Blowing-up method. Robustness of the system. Normal form.
- Bifurcation. Hopf-Takens bifurcation. Examples of higher-dimensional systems.
- Types of partial differential equations. Classification of partial differential equations of the second order. Heat conduction equation. Nonstacionary heat conduction. Fouriers method.
- Method of decomposition by eigenfunctions. Application of Laplace transform to the equation of parabolic type.
- Wave equation. Equation of oscillation of a string. D´ Alambert formula. Application of Fouriers method of separation of variables to the wave equation.
- Laplacian in Cartesian, cylindrical and spherical coordinates. Laplace equation. Boundary-value problems for Laplace equation. Dirichlet problem for disc and ring.
- Final exam