Cod: CM612
Titular curs: Conf. dr. M. Sterpu
Forma de invatamant: cursuri de zi
Ciclul 2 Anul II
Semestrul 2, Curs: 28h, Seminar: 28h
Nr. credite: 8
Specializare: Matematica
Tip disciplina: obligatorie
Categoria formativa: de specialitate
Obiective:
Prezentarea unor notiuni si rezultate fundamentale ale teoriei bifurcatiei
sistemelor dinamice continue si discrete si aplicarea acestora in
modele din biologie si economie.
Continutul cursului:
- Echivalenta topologica a sistemelor dinamice. Clasificare topologica a echilibrelor si punctelor fixe. Bifurcatii si diagrame de bifurcatie. Stabilitate structurala. Forme normale topologice. Varietati centrale. Modele din biologie si economie: Van der Pol, Hodgkin-Huxley, FitzHugh-Nagumo, prada-pradator.
- Bifurcatii de codimensiune 1 ale echilibrelor in sisteme dinamice continue: bifurcatia sa-nod, bifurcatia Hopf.
- Bifurcatii de codimensiune 1 ale punctelor fixe in sisteme dinamice discrete: bifurcatia fald, bifurcatia flip, bifurcatia Neimark-Sacker.
- Bifurcatii ale orbitelor homoclinice si heteroclinice. Teorema Andronov-Leontovich, teoremele lui Shilnikov, integrala Melnikov.
- Bifurcatii de codimensiune doi ale echilibrelor in sisteme dinamice continue: bifurcatia cuspidala, bifurcatia Bautin, bifurcatia Bogdanov-Takens, bifurcatia fald-Hopf, bifurcatia Hopf-Hopf.
- Bifurcatii de codimensiune 2 ale punctelor fixe in sisteme dinamice discrete: bifurcatia cuspidala, bifurcatia flip generalizata, bifurcatia Neimark-Sacker generalizata, rezonante, bifurcatia fald-flip.
- Analiza numerica a bifurcatiilor. Pachetul software XPP.
Discipline anterioare cerute:
- Algebra liniara M1201
- Analiza matematica M1102, M1202
- Ecuatii diferentiale M2305
- Sisteme dinamice CM5xx
Bibliografie:
- Arrowsmith D.K., Place C.M. An introduction to dynamical systems, Cambridge University Press, 1994.
- Chow S.N., Li C., Wang D., Normal forms and bifurcations of planar vector fields, Cambridge Univ. Press, Cambridge, 1994.
- Chow S.N., Hale J., Methods of bifurcation theory, Springer, New-York, 1982.
- Dumortier F., Roussarie R., Sotomayor J., Zoladek, H., Bifurcations of planar vector fields, nilpotent singularities and abelian integrals, Springer, Berlin, 1991.
- Ermentrout B. XPPAUT, http://www.math.pitt.edu
- Ermentrout B. Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students, SIAM, 2002.
- Georgescu A., Moroianu M., Oprea I. Teoria bifurcatiei. Principii si aplicatii, Ed. Univ. Pitesti, 1999.
- Guckenheimer J., Holmes P. Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, New-York, 1983.
- Hale J.K., Kocak H. Dynamics and bifurcations, Springer, New York, 1991.
- Kuznetsov Yu. Elements of applied bifurcation theory, Springer, New York, 2004.
- Murray J.D. Mathematical Biology, Springer, Berlin, 1993.
- Rocsoreanu C., Georgescu A., Giurgiteanu N. The FitzHugh-Nagumo model. Bifurcation and dynamics, Kluwer Academic Publishers, Dordrecht, 2000.
- Rocsoreanu C. Bifurcatiile sistemelor dinamice continue. Aplicatii in economie si biologie, Ed. Universitaria, Craiova, 2006.
- Sterpu M., Rocsoreanu C. Modelarea si simularea proceselor economice, Editura Universitaria, Craiova, 2007.
- Tu P. Dynamical systems. An introduction with applications in economics and biology, Springer, Berlin, 1994.
Fisa disciplinei Teoria bifurcatiilor si aplicatii in biologie (pdf)
|
|
