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

#### Course Description

Dynamical systems. Linear and nonlinear systems. Mathematical modeling. Lyapunov stability theory. Robustness. Bifurcations.

#### 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

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

[FER3-EN] Computing - study

Elective Courses
(6. semester)
[FER3-EN] Electrical Engineering and Information Technology - study

Elective Courses
(6. 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

#### For students

#### General

**ID**223338

Summer semester

**5**ECTS

**L0**English Level

**L1**e-Learning

**60**Lectures

**0**Seminar

**0**Exercises

**0**Laboratory exercises

**0**Project laboratory

#### Grading System

**85**Excellent

**70**Very Good

**55**Good

**45**Acceptable