### Differential Equations and Stability Theory

#### Course Description

#### General Competencies

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.

#### Learning Outcomes

- 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

**Lectures**Lectures.

**Exams**Midexam, exam, seminar

**Consultations**Consulting.

**Seminars**seminar.

#### Grading Method

Continuous Assessment | Exam | |||||
---|---|---|---|---|---|---|

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.
- Exam
- 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

#### Study Programmes

Control Engineering and Automation -> Electrical Engineering and Information Technology (Profile)

Electrical Engineering Systems and Technologies -> Electrical Engineering and Information Technology (Profile)

Electrical Power Engineering -> Electrical Engineering and Information Technology (Profile)

Electronic and Computer Engineering -> Electrical Engineering and Information Technology (Profile)

Electronics -> Electrical Engineering and Information Technology (Profile)

Information Processing -> Information and Communication Technology (Profile)

Telecommunication and Informatics -> Information and Communication Technology (Profile)

Wireless Technologies -> Information and Communication Technology (Profile)

Software Engineering and Information Systems -> Computing (Profile)

Computer Engineering -> Computing (Profile)

Computer Science -> Computing (Profile)

#### Literature

*Differential Equations and Dynamical Systems*, Springer

*Nonlinear Dynamics and Chaos, With Applications to Physics, Biology, Chemistry, and Engineering*, Perseus Books Publishihg

*Partial Differential Equations for Scientists and Engineers Stanley J. Farlow John Wiley & Sons 1982*, Stanley J. Farlow John Wiley & Sons

*Diferencijalne jednadžbe i teorija stabilnosti*, Element

*Fundamentals of Differential Equations and Boundary Value Problems*, Addison-Wesley Publishing Company

#### Lecturers in Charge

#### Grading System

**4**ECTS

**L0**English Level

**L1**e-Learning

**45**Lecturers

**0**Exercises

**0**Laboratory exercises

#### Grading

**80**Excellent

**70**Very Good

**60**Good

**50**Acceptable