### Dynamical Systems, Mathematical Aspect of Stability and Control

#### Learning Outcomes

- Recognize basic notions of dynamical systems theory
- Describe simple systems using dynamical systems theory
- Define Lyapunov stability
- Express statements of basic Lyapunov theorems
- Analyze stability of the system
- Define robustness and bifurcations of system
- Analyze bifurcation by theoretical and numerical methods

#### Forms of Teaching

**Lectures**Lectures

**Seminars and workshops**Seminars held by students as a form of active participation in teaching

**Partial e-learning**Homeworks

#### Grading Method

By decision of the Faculty Council, in the academic year 2019/2020. the midterm exams are cancelled and the points assigned to that component are transferred to the final exam, unless the teachers have reassigned the points and the grading components differently. See the news for each course for information on knowledge rating.

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

Type | Threshold | Percent of Grade | Threshold | Percent of Grade | ||

Homeworks | 0 % | 10 % | 0 % | 0 % | ||

Class participation | 0 % | 10 % | 0 % | 0 % | ||

Mid Term Exam: Written | 0 % | 50 % | 0 % | |||

Final Exam: Written | 0 % | 50 % |

#### Week by Week Schedule

- Motivation for qualitative theory of differential equations- mathematical pendulum, oscillation of electrical circuit; Discrete and continuous dynamical system; Phase space; Equilibrium point; Limit cycle.
- Dissipative and conservative dynamical systems; Autonomous and nonautonomous dynamical systems.
- Classification of phase portraits; Oscillatory and nonoscillatory equilibrium points; Saddle, node, focus, center.
- Definition of Lyapunov stability; Reduction of nonlinear systems; Stable, unstable and central manifold.
- Linearization; Hyperbolic equilibrium point; Hartman-Grobman theorem; Oscillator; Duffing oscillator.
- Bendixson criteria for periodic solutions; Poincare index theory and limit cycles.
- Midterm exam.
- Lyapunov stability; Asymptotic stability; Energy method (potential method); Midterm exam.
- Lyapunov stability theorem for autonomous systems; Lyapunov instability theorem for autonomous systems.
- LaSalle principle and asymptotic stability; Poincare-Bendixson theorem and limit cycles.
- Global and local stability; Asymptotic, uniform and exponential stability; Lyapunov stability theorem.
- Criteria for asymptotic stability; Criteria for uniform stability; Criteria for exponential stability; Robustness.
- Robustness (structural stability) of discrete and continuous dynamical systems; Local bifurcations-saddle-node, transcritical, pitchfork; Period doubling bifurcation; Chaos; Hopf bifurcation, limit cycle, change of stability.
- Nondegenerate and degenerate Hopf bifurcation; Global bifurcations; Homoclinic bifurcation; Bogdanov-Takens bifurcation; Lorenz meteorological system; Strange attractor.
- Final exam; Seminar; Project.

#### Study Programmes

##### University undergraduate

Computing (study)

Elective Courses
(6. semester)
Electrical Engineering and Information Technology (study)

Elective Courses
(6. semester)
##### University graduate

Computer Engineering (profile)

Mathematics and Science
(2. semester)
Computer Science (profile)

Mathematics and Science
(2. semester)
Control Engineering and Automation (profile)

Mathematics and Science
(2. semester)
Electrical Engineering Systems and Technologies (profile)

Mathematics and Science
(2. semester)
Electrical Power Engineering (profile)

Mathematics and Science
(2. semester)
Electronic and Computer Engineering (profile)

Mathematics and Science
(2. semester)
Electronics (profile)

Mathematics and Science
(2. semester)
Information Processing (profile)

Mathematics and Science
(2. semester)
Software Engineering and Information Systems (profile)

Mathematics and Science
(2. semester)
Telecommunication and Informatics (profile)

Mathematics and Science
(2. semester)
Wireless Technologies (profile)

Mathematics and Science
(2. semester)
#### Literature

(.),

*Luka Korkut, Vesna Županović, Diferencijalne jednadžbe i teorija stabilnosti; Element; 2009; ISBN: ISBN 978-953-197-559-9*,
(.),

*Shankar Sastry, Nonlinear Systems Analysis, Stability, and Control, Springer-Verlag 1999, ISBN 978-1-4757-3108-8*,
(.),

*Steven H. Strogatz, Nonlinear Dynamics and Chaos, With Applications to Physics, Biology, Chemistry, and Engineering; Perseus Books Publishihg; 2000; ISBN: 0738204536, 9780738204536*,#### Lecturers

#### General

**ID**183493

Summer semester

**5**ECTS

**L0**English Level

**L1**e-Learning

**60**Lectures

**0**Exercises

**0**Laboratory exercises

**0**Project laboratory

#### Grading System

**85**Excellent

**70**Very Good

**55**Good

**45**Acceptable