Centro di Ricerca Matematica "Ennio De Giorgi", Pisa
Conservatorio di Santa Chiara, San Miniato (Pisa), Italy
The “Summer School of Mathematics for Economics and Social Sciences” aims to improve the knowledge of mathematical methods among graduate students in economics and social sciences, with a focus on those techniques which albeit widespread in use are not properly covered in typical graduate programmes. The School is an interdisciplinary venue intended to foster the interaction of people coming from the too often separated communities of mathematical and social scientists. It is organized by the Mathematics Research Center “Ennio De Giorgi” and supported by the PhD programme in Economics of the Scuola Superiore Sant'Anna.
Topics: Structural stability; Codimension one bifurcations of vector fields; Codimension one bifurcations of maps; Introduction to codimension two bifurcations
Lecturer: Florian Wagner, CeNDEF, University of Amsterdam
Participation subject to selection. Only 20-25 positions available. For applications link to http://www.crm.sns.it/event/256/financial.html or write to crm@crm.sns.it..
Deadline for application: 5 August, 2012
Topics considered prerequisite for the course: complex numbers, eigenvectors, eigenvalues and diagonalisation, Taylor’s theorem, implicit function theorem, linear ODEs and one-dimensional autonomous nonlinear ODEs.
Scientific Committee: Giulio Bottazzi, Giorgio Fagiolo (Scuola Superiore Sant'Anna), Davide Fiaschi (Università di Pisa), Stefano Marmi (Scuola Normale Superiore)
Syllabus of the course
Introduction
- Review of linear theory
Structural stability
- Equivalence classes of linear dynamics
- Differentiable and topological equivalence
- Hartman-Grobman theorem
- Structural stability
- General notion of bifurcation
Codimension one bifurcations of vector fields
- Saddle-node bifurcation
- Normal forms
- Hopf bifurcation
- Invariant manifolds
- Homoclinic and heteroclinic bifurcations
- Symmetry
- Pitchfork bifurcation
Codimension one bifurcations of maps
- Poincaré section
- Saddle-node bifurcation
- Neimark-Sacker bifurcation
- Resonances
- Homoclinic tangencies
- Horseshoes
Introduction to Codimension two bifurcations