# Computing

## Dynamical Systems, Mathematical Aspect of Stability and Control

- Class 60
- Practice 0
- Independent work 90

### Course title

Dynamical Systems, Mathematical Aspect of Stability and Control

### Lecture type

Elective

### Course code

183493

### Semester

6

### ECTS

5

### Lecturers and associates

### Course objectives

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.

### Required reading

(.), 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 Publishing; 2000; ISBN: 0738204536, 9780738204536,

#### Online education during epidemiological measures

- Study program duration
- 6 semesters (3 years)
- Semester duration
- 15 weeks of active teaching + 5 examination weeks
- Total number of ECTS points
- 180
- Title
- Bacc.ing.comp (Bachelor of Science in Computing)

**Academic calendar**

#### Minimal 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