syllabus
1. Information regarding the programme
1.1 Higher education institution 	Babes-Bolyai University	
1.2 Faculty	Faculty of Mathematics and Computer Science	
1.3 Department	Department of Mathematics	
1.4 Field of study	Computer Science	
1.5 Study cycle	Bachelor	
1.6 Study programme / Qualification 	Computer Science	

2. Information regarding the discipline 
2.1 Name of the discipline	Dynamical Systems	
2.2 Course coordinator 	Conf. dr. Adriana Buica	
2.3 Seminar coordinator	Conf. dr. Adriana Buica	
2.4. Year of study	1	2.5 Semester	2	2.6. Type of evaluation	E	2.7 Type of discipline	compulsory	

3. Total estimated time (hours/semester of didactic activities) 
3.1 Hours per week 	4	Of which: 3.2 course	2	3.3 seminar/laboratory	1/1	
3.4 Total hours in the curriculum 	56	Of which: 3.5 course	28	3.6 seminar/laboratory	14/14	
Time allotment:	hours	
Learning using manual, course support, bibliography, course notes	40	
Additional documentation (in libraries, on electronic platforms, field documentation) 	20	
Preparation for seminars/labs, homework, papers, portfolios and essays	35	
Tutorship	5	
Evaluations	20	
Other activities: ..................		
3.7 Total individual study hours 	120	
3.8 Total hours per semester	176	
3.9 Number of ECTS credits	6	

4. Prerequisites (if necessary)
4.1. curriculum	•	Mathematical Analysis 	
4.2. competencies	•	Calculate derivatives and integrals for real functions	

5. Conditions (if necessary)
5.1. for the course	•		
5.2.  for the seminar /lab activities	Computers for the laboratory acitvity	

6. Specific competencies acquired 
Professional competencies	•	Understanding of basic concepts of dynamical systems theory  and use them to problem-solving activities
•	Ability to understand and approach problems of modeling nature from other sciences	
Transversal competencies	Ability to work independently and/or in a team in order to solve problems in defined professional contexts	

7. Objectives of the discipline (outcome of the acquired competencies)
7.1 General objective of the discipline
	•	Introduction to the basic problems of dynamical systems theory as well as the discussion of some related mathematical models 	
7.2 Specific objective of the discipline
	•	To understand the concepts of equilibrium point, orbit, periodic orbit, stability, chaos and to operate with them at least in some simple situations.  	

8. Content
8.1 Course	Teaching methods	Remarks	
1.	Introduction to differential equations (notions, initial conditions, boundary conditions, examples, fundamental problems)	Exposure: description, explanation, examples		
2.	Linear differential equations (existence and uniqueness theorem, fundamental theorems, the variation of constants method)	Exposure: description, explanation, examples		
3.	Linear differential equations with constant coefficients. Applications: Galileo’s pendulum. Harmonic oscillators.	Exposure: description, explanation, examples		
4.	Linear differential equations with variable coefficients (Euler equations, the power series method)	Exposure: description, explanation, examples		
5.	Linear differential systems (fundamental theorems, exponential of a matrix, reduction to a higher order differential equation).	Exposure: description, explanation, examples		
6.	First order one-dimensional nonlinear differential equations (Peano theorem, local and global existence and uniqueness theorem, non-uniqueness, blow-up)	Exposure: description, explanation, examples		
7.	Direction field. First integrals. Scalar differential equations vs. Pfaff equations vs. planar autonomous systems	Exposure: description, explanation, examples		
8.	Few classes of integrable differential equations (separable, Bernoulli, Ricatti, homogeneous). Picard sequence of successive approximations	Exposure: description, explanation, examples		
9.	Numerical methods for first order one-dimensional nonlinear differential equations (Euler algorithm, Runge-Kutta algorithm).	Exposure: description, explanation, examples		
10.	Higher dimensional nonlinear differential equations (existence and uniqueness theorems, successive approximations, numerical methods).	Exposure: description, explanation, examples		
11.	Autonomous nonlinear differential equations. Flow, orbit, phase portrait, singular points, cycles.	Exposure: description, explanation, examples		
12.	Classification of singular points. Stability of singular points of linear systems and of nonlinear systems through linearization	Exposure: description, explanation, examples		
13.	Discrete dynamical systems (notions, examples, fundamental problems, numerical methods vs. discrete equations).	Exposure: description, explanation, examples		
14.	Linear discrete equations. The logistic equation.	Exposure: description, explanation, examples		
Bibliography
1.	P. Blanchard, R.L. Devaney, G.R. Hall, Differential Equations, Brooks/Cole, Cengage Learning, 2012. 
2.	S. Lynch, Dynamical systems with applications using MAPLE, Birkhauser, 2001. 
3.	Gh. Micula, P. Pavel, Ecuatii diferentiale si integrale prin probleme si exercitii, Ed. Dacia, Cluj-Napoca,1989.
4.	E.R. Scheinerman, Invitation to Dynamical Systems,  internet version freely available at http://www.ams.jhu.edu/~ers/invite.html
5.	Jon H. Davis, Differential Equations with MAPLE: an Interactive Approach, Birkhäuser, 2001	
8.2 Seminar / laboratory	Teaching methods	Remarks	
1.	Seminar 1. Linear homogeneous differential equations with constant coefficients. General solutions and properties of solutions (periodicity, oscillations, boundedness…)	Explanation, dialogue, examples		
2.	Laboratory 1. Introduction to Maple. Basic notions.	Explanation, dialogue, examples		
3.	Seminar 2. Linear differential equations (the method of undetermined coefficients, the Lagrange method).	Explanation, dialogue, examples		
4.	Laboratory 2. The use of Maple to find the general solution of linear differential equations, to solve initial and boundary value problems and to study the properties of solutions	Explanation, dialogue, examples		
5.	Seminar 3. Euler equations. Power series method. Linear differential systems.	Explanation, dialogue, examples		
6.	Laboratory 3. The use of Maple to find the general solution to Euler equations and to linear systems. Power series method. 	Explanation, dialogue, examples		
7.	Seminar 4. Separable equations. Direction field. Other classes of integrable differential equations	Explanation, dialogue, examples		
8.	Laboratory 4. Integrable differential equations. Direction fields	Explanation, dialogue, examples		
9.	Seminar 5. Numerical methods.	Explanation, dialogue, examples		
10.	Laboratory 5. Numerical methods	Explanation, dialogue, examples		
11.	 Seminar 6. Phase portraits. Singular points and their stability.	Explanation, dialogue, examples		
12.	Laboratory 6. Phase portraits	Explanation, dialogue, examples		
13.	 Seminar 7. Discrete equations	Explanation, dialogue, examples		
14.	Laboratory 7. Discrete equations	Explanation, dialogue, examples		
Bibliography
1.	P. Blanchard, R.L. Devaney, G.R. Hall, Differential Equations, Brooks/Cole, Cengage Learning, 2012. 
2.	S. Lynch, Dynamical systems with applications using MAPLE, Birkhauser, 2001. 
3.	Gh. Micula, P. Pavel, Ecuatii diferentiale si integrale prin probleme si exercitii, Ed. Dacia, Cluj-Napoca,1989.
4.	E.R. Scheinerman, Invitation to Dynamical Systems,  internet version freely available at http://www.ams.jhu.edu/~ers/invite.html
5.	Jon H. Davis, Differential Equations with MAPLE: an Interactive Approach, Birkhäuser, 2001		

9. Corroborating the content of the discipline with the expectations of the epistemic community, professional associations and representative employers within the field of the program

•	 The content of this discipline is synchronized with the curriculum of most of the important universities from our country and from abroad where the applied mathematics plays an important role.  	

10. Evaluation
Type of activity	10.1 Evaluation criteria	10.2 Evaluation methods	10.3 Share in the grade (%)	
10.4 Course	To know the notions and their properties by examples or counterexamples. To be able to apply the theoretical results in concrete problems.	Exam	65%	
				
10.5 Seminar/lab activities	Solving problems skills	Two tests	20%	
	Interest and implication in each lab activity. One final test in the last lab.	Dialogue in each lab and one final test	15%	
10.6 Minimum performance standards	
•	Presence at least at 90% from the lab activities, at least 10% points from the lab activity, at least 15% points from the written final exam and the minimum passing grade is 5.	

Date				Signature of course coordinator 	Signature of seminar coordinator
14-10-2012		             Conf. dr. Adriana Buica	            	Conf. dr. Adriana Buica
Date of approval  					  Signature of the head of department	
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