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., abuica@math.ubbcluj.ro
Lect. ANDRAS Szilard Karoly, andrasz@math.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.
	
Bibliography	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%