Dynamical Systems (MATH 307)

2022 Fall
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
6.00 / 6.00 ECTS (for students admitted in the 2013-14 Academic Year or following years)
Nilay Duruk Mutluba┼č -nilaydm@sabanciuniv.edu,
English
Undergraduate
--
Formal lecture,Interactive lecture,Seminar
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CONTENT

Qualitative theory of ordinary differential equations (ODEs). Existence and uniqueness, geometrical representation of ODEs. Construction of phase portraits. Nonlinear systems, local and global behavior, the linearization theorem. Periodic orbits and limit sets, Poincare-Bendixson theory. The stable manifold theorem, homoclinic and heteroclinic points. Bifurcation diagrams. State reconstruction from data, embedding.

OBJECTIVE

To teach the fundamental theory of ODE's; the fundametals of dynamical systems and their connections with problems from a wide variety of areas.

LEARNING OUTCOME

the students should be able to
(a) Draw phase portraits in the plane
(b) Construct proofs for existence and uniqueness for linear systems
(c) Identify fixed/equilibrium points, determine their stability, and analyze local and global behavior
(d) Rule out closed orbits for gradient systems and construct Liapunov functions
(e) Utilize theorems to establish the existence of closed orbits, such as Poincare-Bendixson theorem
(f) Study in detail Lienard systems (or those can be converted to a Lienard systems)
(g) Employ perturbation theory for weakly nonlinear oscillators
(h) Create bifurcation diagrams for saddle-node, transcritical, and pitchfork bifurcations
(i) Investigate fixed points and their stability for one-dimensional maps

ASSESSMENT METHODS and CRITERIA

  Percentage (%)
Final 40
Midterm 30
Homework 30

RECOMENDED or REQUIRED READINGS

Readings

Arrowsmith D.K., Place C.M., Dynamical Systems- Differential Equations, maps and chaotic behaviour, Chapman&Hall, 1992.