Course Details for A.Y. 2016/2017

The information on this page refer to the academic year 2016/2017. Therefore, several links are not available and data may be incomplete. To read the current information on the course, if it is still available, go to the university course catalogue . To read information relating other academic years, use the list at the bottom of this page.

A. Year:

2016/2017

Name:

Dynamical systems and bifurcation theory / Dynamical systems and bifurcation theory

Type:

Course

Basic information

Code:

I0459

Sector:

MAT/05

Credits:

: Master Degree in Mathematical Engineering 6 CFU (b)

Term:

1^st semester

Degree(s):

Master Degree in Mathematical Engineering 1^st anno curriculum Comune Compulsory

Language:

English

Teacher(s):

Bruno Rubino

Same as:

Dynamical systems and bifurcation theory

Course Objectives

To render the students able to solve practical problems of bifurcation analysis. To stimulate the emergence of an engineering critical sense in using algorithms of Applied Mathematics finalized to system design.

Course Content

LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS. Exponential of a linear operator, fundamental matrix. Stability for linear systems: stable, unstable and center subspaces. Sinks and sources. Classification of the equilibrium point for a planar linear system: saddles, nodes, foci and centers. Phase portrait in dimension two. Non homogeneous linear systems.
LOCAL THEORY FOR NON LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS. Invariant sets. Hyperbolic equilibrium points. Linearization. Stable manifold theorem. Center manifold theorem. TheHartman-Grobman theorem. Hyperbolic equilibrium points for planar systems: saddles, nodes, foci and centers. Stability and Liapunov function. Non hyperbolic equilibrium points for a planar system. Center manifold theory. Normal form theory. Gradient and Hamiltonian systems.
GLOBAL THEORY FOR NON LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS. Dynamical systems. Limits sets and attractors. Periodic orbits, limit cycles and separatrix cycles. Homoclinic and heteroclinic orbits. Compound separatrix cycles. Dulac theorem. Poincaré map for a cycle. Poincaré map for a focus. Planar case and Poincaré map. The Stable manifold theorem fo periodic orbits. Floquet theorem. The Center manifold theorem for periodic orbits. Liouvile theorem for the fundamental matrix. Characteristic exponents and multipliers for a period orbit.

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

be able to calculate the exponential of a matrix;
know the notion of stability for an equilibrium point and for a limit cycle;
be able to solve practical problems of bifurcation analysis.

Prerequisites and Learning Activities

Linear Algebra, basic theory of Ordinary differential equations and basic of continuum mechanics.

Assessment Methods and Criteria

Written and oral.

Textbooks

Lawrence Perko, Differential Equations and Dynamical Systems , Springer. 2001. Third edition (ISBN: 0-387-9516-4)

Notes

Si veda anche la pagina sul vecchio sito di ingegneria: http://www.ing.univaq.it/cdl/scheda_corso.php?codice=I2W054

Course page updates

This course page is available (with possible updates) also for the following academic years:

To read the current information on this course, if it is still available, go to the university course catalogue .

Course information last updated on: 05 maggio 2016, 17:30