Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MME1009 | Simulation of Continuous Systems |
| Teaching Staff in Charge |
Assoc.Prof. BUICA Adriana, Ph.D., abuica math.ubbcluj.ro |
| Aims |
|
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. |
| Content |
|
1. One dimensional flows. Examples of bifurcations.
2. Bifurcations in scalar autonomous differential equations. 3. Euler’s 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. |
| References |
|
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. |
| Assessment |
|
Final grade consists from:
- Final written exam: 70% - Activity during the semester (laboratory works and seminar works): 30% |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |