Study

Computing

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

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,

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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