Course Code TAE463
Semester 7
Category Optional
Points 3
ECTS Units 5
Recommended Reading

Dynamical Systems with Applications using Matlab, S. Lynch, Birkhauser 2014.
Differential Equations, Dynamical Systems and an Introduction to Chaos, M. Hirsch, S. Smale, R. Devaney, Elsevier Academic Press, 2004.
Differential Equations and Dynamical Systems, L. Perko, Springer, 2000.
Dynamics and Bifurcations, J. Hale, H. Kocak, Springer-Verlag, 1991.
Nonlinear Oscilations, Dynamical Systems and Bifurcations of Vector Fields, J. Guckenheimer, P. Holmes, Springer,1983.
Chaos, An Introduction to Dynamical Systems, K. Alligoog, T. Sauer, J. Yorke, Springer, 1997

Course Description

1. Autonomous Differential Equations of First-order

  • Critical points, stability, linear stability analysis, existence and uniqueness, bifurcations

2. Autonomous Systems on the plane

  • Linear Systems: classification, stable and unstable manifolds, phase diagrams
  • Non-Linear Systems: topological equivalence, critical points and linearization, phase diagrams
  • Limit cycles: existence and uniqueness, rule-out limit cycles
  • Bifurcations: saddle-node, transcritical, pitchfork, Hopf
  • Hamiltonian Systems, Gradient Systems, Reversible Systems

3. Poincare maps and non-autonomous systems on the plane

4. Three-Dimensional Autonomous Systems and Chaos

  • Linear and non-linear systems: critical points, stability, phase diagrams
  • Lorenz equations: properties, critical points, asymptotic stability, strange tractors, chaos

5. Discrete Dynamic Systems

  • Linear and nonlinear discrete systems: fixed points, stability, cobwebs, periodic solutions, trajectories, period doubling sequences
  • Triangular map
  • Logistic map and the Feigenbaum constant

6. Complexity

  • Complex iterations
  • Fractals
  • Networks