Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master

SUBJECT

Code
Subject
MME1006 Nonlinear Dynamic Systems
Section
Semester
Hours: C+S+L
Category
Type
Applied Mathematics
4
2+2+0
speciality
optional
Teaching Staff in Charge
Assoc.Prof. BUICA Adriana, Ph.D.,  abuicamath.ubbcluj.ro
Lect. ANDRAS Szilard Karoly,  andraszmath.ubbcluj.ro
Aims
The aim of this course is to provide students with the problems and techniques in the study of trajectory behavior of nonlinear dynamical systems both continuous and discrete. We will study the geometry, stability and bifurcations of equilibria and periodic solutions.
Content
1. One-dimensional flows. Example of bifurcations.
2. Bifurcations in scalar autonomous differential equations.
3. Bifurcation of scalar maps. The logistic map.
4. Geometry and stability of periodic solutions of scalar nonautonomous equations.
5. Bifurcation of periodic solutions of scalar nonautonomous equations.
6. General properties of planar autonomous systems.
7. Examples of elementary bifurcations in planar autonomous systems.
8. Bifurcations in linear planar systems.
9. The behavior of the orbits near hyperbolic equilibria.
10. The behavior of the orbits near equilibria with a zero eigenvalue.
11. The behavior of the orbits near equilibria with purely imaginary eigenvalues.
12. Existence, stability and bifurcations of periodic orbits.
13. Structurally stable vector fields.
14. Few examples in higher dimensions.
References
1. V. Barbu, Ecuatii diferentiale, Editura Junimea, Iasi, 1985.
2. F. Diacu, An Introduction to Differential Equations. Order and Chaos, W.H. Freeman and Company New York, 2000.
3. J. Hale, H. Kocak, Dynamics and Bifurcations, Springer Verlag New York Inc., 1991.
4. M.W. Hirsch, S. Smale and R.L. Devaney, Differential equations, dynamical systems, and an introduction to chaos, Elsevier Academic Press, 2004.
5. H. Khalil, Nonlinear Systems, Prentince Hall Inc., 1996.
6. I.A. Rus, Ecuatii diferentiale, ecuatii integrale si sisteme dinamice, Transilvania Press, 1996.
7. D. Trif, Metode numerice pentru ecuatii diferentiale si sisteme dinamice, Transilvania Press, 1997.
Assessment
Final grade consists from:
- Final written exam: 70%
- Activity during the semester (test, reports ): 30%
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject