Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Informatica
Programul de studii: Modelare si simulare - în limba engleză
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MME1009	Simulation of Continuous Systems	

Semester	Hours: C+S+L	Category	Status	
2	2+1+0	speciality	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
BUICA Adriana	Ph.D.	Assoc.Prof.	Applied Mathematics	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
 The goal of this course is to provide students with an introduction to problems and techniques of studying and computing the trajectory behavior of dynamical systems that are modeled through both ordinary differential equations and difference equations. We will focus on the geometry, stability and bifurcation of equilibrium points and periodic solutions. The simulations will be made using either Maple or Matlab.	

IV. Course contents
1. One dimensional flows. Examples of bifurcations. 
2. Bifurcations in scalar autonomous differential equations. 
3. Euler algorithm and maps. 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.

V. Bibliography
1. V. Barbu, Ecuatii diferentiale, Editura Junimea, Iasi, 1985. 
2. F.E. Cellier and E. Kofman, Continuous System Simulation, Springer, New York, 2006. 
3. F. Diacu, An Introduction to Differential Equations. Order and Chaos, W.H. Freeman and Company New York, 2000.  
4. J. Hale, H. Kocak, Dynamics and Bifurcations, Springer Verlag New York Inc., 1991.
5. M.W. Hirsch, S. Smale and R.L. Devaney, Differential equations, dynamical systems, and an introduction to chaos, Elsevier Academic Press, 2004.
6. H. Khalil, Nonlinear Systems, Prentince Hall Inc., 1996. 
7. S. Lynch, Dynamical systems with applications using MAPLE, Birkhauser, 2001.
8. I.A. Rus, Ecuatii diferentiale, ecuatii integrale si sisteme dinamice, Transilvania Press, 1996.
9. D. Trif, Metode numerice pentru ecuatii diferentiale si sisteme dinamice, Transilvania Press, 1997.

VI. Thematic of didactic activities per weeks
Lecture 1. One dimensional flows. Examples of bifurcations. [4,5]
Lecture 2. Bifurcations in scalar autonomous differential equations. [4,5]
Lecture 3. Euler’s algorithm and maps. Bifurcation of scalar maps. The logistic map. [4,9]
Lecture 4. Geometry and stability of periodic solutions of scalar nonautonomous equations. [4,5,6]
Lecture 5.  Bifurcation of periodic solutions of scalar nonautonomous equations. [4,5]
Lecture 6.  General properties of planar autonomous systems. [3,4,5,6]
Lecture 7.  Examples of elementary bifurcations in planar autonomous systems. [4,5]
Lecture 8.  Bifurcations in linear planar systems. [4]	
Lecture 9.  The behavior of the orbits near hyperbolic equilibria. [3,4,5,6,8]
Lecture 10. The behavior of the orbits near equilibria with a zero eigenvalue. [3,4,5,6]	
Lecture 11. The behavior of the orbits near equilibria with purely imaginary eigenvalues. [3,4,5,6]
Lecture 12.  Existence, stability and bifurcations of periodic orbits. [1,4,5,6]   
Lecture 13.  Structurally stable vector fields. [4]
Lecture 14.  Few examples in higher dimensions. [1,4,8]

Seminar 1. Bifurcations in scalar autonomous differential equations. [4,5]
Seminar 2. Bifurcation of scalar maps. [4,5]
Seminar 3. Periodic solutions of scalar nonautonomous equations. [4,5,6] 
Seminar 4. Bifurcations in planar autonomous systems. [4,5]
Seminar 5. The behavior of the orbits near hyperbolic equilibria. [3,4,5,6]
Seminar 6. The behavior of the orbits near nonhyperbolic equilibria. [3,4,5,6]
Seminar 7. Existence, stability and bifurcations of periodic orbits. [4,5,6]   

VII. Didactic methods used
Lectures, interactive presentations, exercises, individual study, homework assignments.	

VIII. Assessment
Final grade consists from:
- Final written exam: 70%
- Activity during the semester: 30%

IX. Additional references
1. R.L. Borrelli, C. Coleman, W.E. Boyce, Differential Equations. Laboratory Workbook, John Wiley & Sons Inc., 1992.
 2. D.N. Burghes, M.S. Borrie, Modelling with Differential Equations, Ellis Horwood, 1981.
 3. A. Halanay, Teoria calitativa a ecuatiilor diferentiale, Ed. Academiei R.P.R., 1963. 
 4.  Gh. Micula, P. Pavel, Ecuatii diferentiale si integrale prin probleme si exercitii, Ed. Dacia, Cluj-Napoca, 1989.
 5. Gh. Morosanu, Ecuatii diferentiale. Aplicatii, Ed. Academiei R.S.R., Bucuresti, 1989.
 6. R. Radheffer, Differential Equations. Theory and Applications, Jones and Bartlett Publishers, 1991.	

	Dean,							Chair head,

	Date,							Course responsible,
								Assoc.Prof. BUICA Adriana, Ph.D.