Popis predmeta

Course Description

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

Learning Outcomes

  1. Recognize basic notions of dynamical systems theory
  2. Describe simple systems using dynamical systems theory
  3. Define Lyapunov stability
  4. Express statements of basic Lyapunov theorems
  5. Analyze stability of the system
  6. Define robustness and bifurcations of system
  7. 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

  1. Motivation for qualitative theory of differential equations- mathematical pendulum, oscillation of electrical circuit, Discrete and continuous dynamical system; Phase space; Equilibrium point; Limit cycle
  2. Dissipative and conservative dynamical systems, Autonomous and nonautonomous dynamical systems
  3. Classification of phase portraits; Oscillatory and nonoscillatory equilibrium points, Saddle, node, focus, center
  4. Definition of Lyapunov stability, Reduction of nonlinear systems; Stable, unstable and central manifold
  5. Linearization; Hyperbolic equilibrium point; Hartman-Grobman theorem, Oscillator; Duffing oscillator
  6. Bendixson criteria for periodic solutions, Poincare index theory and limit cycles
  7. Midterm exam
  8. Lyapunov stability; Asymptotic stability, Energy method (potential method), Midterm exam
  9. Lyapunov stability theorem for autonomous systems, Lyapunov instability theorem for autonomous systems
  10. LaSalle principle and asymptotic stability, Poincare-Bendixson theorem and limit cycles
  11. Global and local stability, Asymptotic, uniform and exponential stability, Lyapunov stability theorem
  12. Criteria for asymptotic stability, Criteria for uniform stability, Criteria for exponential stability; Robustness
  13. 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
  14. Nondegenerate and degenerate Hopf bifurcation, Global bifurcations; Homoclinic bifurcation; Bogdanov-Takens bifurcation, Lorenz meteorological system; Strange attractor
  15. 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
Audio Technologies and Electroacoustics (profile)
Free Elective Courses (2. semester)
Communication and Space Technologies (profile)
Free Elective Courses (2. semester)
Computational Modelling in Engineering (profile)
Free Elective Courses (2. semester)
Computer Engineering (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Computer Science (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Control Engineering and Automation (profile)
Mathematics and Science (2. semester)
Control Systems and Robotics (profile)
Free Elective Courses (2. semester)
Data Science (profile)
Free Elective Courses (2. semester)
Electrical Engineering Systems and Technologies (profile)
Mathematics and Science (2. semester)
Electrical Power Engineering (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Electric Machines, Drives and Automation (profile)
Free Elective Courses (2. semester)
Electronic and Computer Engineering (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Electronics (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Information and Communication Engineering (profile)
Free Elective Courses (2. semester)
Information Processing (profile)
Mathematics and Science (2. semester)
Network Science (profile)
Free Elective Courses (2. semester)
Software Engineering and Information Systems (profile)
Free Elective Courses (2. semester) 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,

For students

General

ID 183493
  Summer semester
5 ECTS
L0 English Level
L1 e-Learning
60 Lectures

Grading System

85 Excellent
70 Very Good
55 Good
45 Acceptable

Learning Outcomes

  1. Recognize basic notions of dynamical systems theory
  2. Describe simple systems using dynamical systems theory
  3. Define Lyapunov stability
  4. Express statements of basic Lyapunov theorems
  5. Analyze stability of the system
  6. Define robustness and bifurcations of system
  7. 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

  1. Motivation for qualitative theory of differential equations- mathematical pendulum, oscillation of electrical circuit, Discrete and continuous dynamical system; Phase space; Equilibrium point; Limit cycle
  2. Dissipative and conservative dynamical systems, Autonomous and nonautonomous dynamical systems
  3. Classification of phase portraits; Oscillatory and nonoscillatory equilibrium points, Saddle, node, focus, center
  4. Definition of Lyapunov stability, Reduction of nonlinear systems; Stable, unstable and central manifold
  5. Linearization; Hyperbolic equilibrium point; Hartman-Grobman theorem, Oscillator; Duffing oscillator
  6. Bendixson criteria for periodic solutions, Poincare index theory and limit cycles
  7. Midterm exam
  8. Lyapunov stability; Asymptotic stability, Energy method (potential method), Midterm exam
  9. Lyapunov stability theorem for autonomous systems, Lyapunov instability theorem for autonomous systems
  10. LaSalle principle and asymptotic stability, Poincare-Bendixson theorem and limit cycles
  11. Global and local stability, Asymptotic, uniform and exponential stability, Lyapunov stability theorem
  12. Criteria for asymptotic stability, Criteria for uniform stability, Criteria for exponential stability; Robustness
  13. 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
  14. Nondegenerate and degenerate Hopf bifurcation, Global bifurcations; Homoclinic bifurcation; Bogdanov-Takens bifurcation, Lorenz meteorological system; Strange attractor
  15. 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
Audio Technologies and Electroacoustics (profile)
Free Elective Courses (2. semester)
Communication and Space Technologies (profile)
Free Elective Courses (2. semester)
Computational Modelling in Engineering (profile)
Free Elective Courses (2. semester)
Computer Engineering (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Computer Science (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Control Engineering and Automation (profile)
Mathematics and Science (2. semester)
Control Systems and Robotics (profile)
Free Elective Courses (2. semester)
Data Science (profile)
Free Elective Courses (2. semester)
Electrical Engineering Systems and Technologies (profile)
Mathematics and Science (2. semester)
Electrical Power Engineering (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Electric Machines, Drives and Automation (profile)
Free Elective Courses (2. semester)
Electronic and Computer Engineering (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Electronics (profile)
Free Elective Courses (2. semester) Mathematics and Science (2. semester)
Information and Communication Engineering (profile)
Free Elective Courses (2. semester)
Information Processing (profile)
Mathematics and Science (2. semester)
Network Science (profile)
Free Elective Courses (2. semester)
Software Engineering and Information Systems (profile)
Free Elective Courses (2. semester) 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,

For students

General

ID 183493
  Summer semester
5 ECTS
L0 English Level
L1 e-Learning
60 Lectures

Grading System

85 Excellent
70 Very Good
55 Good
45 Acceptable