# Course Details

#### Name:

**Dynamical systems and bifurcation theory / Dynamical systems and bifurcation theory**

### Basic information

##### Credits:

*Master Degree in Mathematical Engineering:* 6 Ects (b)

##### Degree(s):

Compulsory 1^{st} year Master Degree in Mathematical Engineering curriculum Comune

##### Language:

English

### 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 understand the general concept of a dynamical system, and the significance of dynamical systems for modelling real world phenomena;
- be able to analyze simple dynamical systems to find and classify regular behaviour;
- be able to choose and use methods appropriate to describe, qualitatively and quantitatively, the solution sets of ordinary differential;
- be able to analyse simple dynamical systems to find and classify regular behaviour, sketch phase portraits and determine stable and unstable manifolds;
- be able to determine the existence and stability of periodic orbits in two dimensions using Dulac's criterion, Lyapunov functions, the Poincaré-Bendixson Theorem and Floquet theory;
- be able to use the Centre Manifold Theorem to reduce the order of a system appropriately and bring the reduced system into normal form;
- be able to identify and classify codimension one bifurcations of fixed points and sketch bifurcation diagrams;
- be able to use bifurcation theory to qualitatively describe how dynamical systems taken from applications in science and technology depend on a parameter;
- be familiar with some of the simpler bifurcation scenarios, and how they can be analyzed.

### Prerequisites and Learning Activities

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

### Teaching Methods

**Language**: English

Lectures and exercises. The module is split into two parts:
Dynamical Systems (Prof. Rubino, 4 ECTS) and Bifurcation Theory (Prof. Luongo, 2 ECTS)

### Assessment Methods and Criteria

Written and oral.

### Textbooks

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

### Course page updates

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

*Course information last updated on: 13 giugno 2018, 22:23*