Dynamical Systems, Mathematical Aspect of Stability and Control

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



Seminars and workshops

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

Partial e-learning


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.

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(.), 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,


ID 183493
  Summer semester
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