Dynamical Systems, Mathematical Aspect of Stability and Control

  • Class 60
  • Practice 0
  • Independent work 90
Total 150

Course title

Dynamical Systems, Mathematical Aspect of Stability and Control

Lecture type


Course code






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,

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
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