# Differential equations and dynamical systems

Course Code : | 1000WETDDS |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 1st semester |

Sequentiality: | - |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | Dutch |

Exam period: | exam in the 1st semester |

Lecturer(s) | Sonja Hohloch Yannick Gullentops |

### 1. Prerequisites *

an active knowledge of

- Dutch

- English

Standard facts of multivariable calculus.

### 2. Learning outcomes *

- Basic definitions (ordinary differential equation, systems, ordinary differential equations of higher order).
- Explicit techniques (autonomous differential equations, separation of variables, change of variables, homogeneous differential equations, fractions).
- Existence and uniqueness (fixed point theorem of Banach, theorem of Picard-LindelĂ¶f, theorem of Peano, smooth dependence on initial conditions).
- Linear systems (homogeneous and inhomogeneous, constant coefficients, classification in 2 dimensions).
- Boundary value problems (boundary value problem of Sturm-Liouville, uniqueness, construction of solutions).
- Basic properties of dynamical systems (discrete and continuous dynamical systems; vector fields; flows; fixed points, periodic and non-periodic orbits, commuting flows).
- Local and global behaviour of dynamical systems (stability; invariant sets; local behaviour close to hyperbolic fixed points: the theorem of Hartman-Grobman; limit sets; dynamics in 2 dimensions: the theorem of PoincarĂ©-Bendixon).
- Chaos: Strange attractors (what is chaos; the Lorenz system; strange attractors).

### 3. Course contents *

**Chapter 1: **Basic definitions

- Ordinary differential equations.
- Systems of ordinary differential equations.
- Ordinary differential equations of higher order.

**Chapter 2:** Explicit techniques

- Autonomous ordinary differential equations.
- Separation of variables.
- Change of variables
- Homogeneous differential equations.
- Fractions.

**Chapter 3: **Existence and uniqueness

- Fixed point theorem of Banach
- Theorem of Picard-Lindelöf and the Theorem of Peano.
- Continuous/smooth dependence on initial conditions.

**Chapter 4: **Linear systems

- Homogeneous and inhomogeneous systems.
- Systems with constant coefficients.
- Classification in 2 dimensions.

**Chapter 5:** Boundary value problems

- Boundary value problem of Sturm-Liouville.
- Uniqueness.
- Construction of solutions.

**Hoofdstuk 6: **Basic properties of dynamical systems

- Discrete and continuous dynamical systems.
- Vector fields and flows.
- Fixed points, periodic and non-periodic orbits.
- Commuting flows.

**Chapter 7: **Local and global behaviour of dynamical systems

- Stability and invariant sets.
- Local behaviour close to hyperbolic fixed points: the theorem of Hartman-Grobman.
- Limit sets.
- Dynamics in 2 dimensions: the theorem of Poincaré-Bendixon.

**Chapter 8:** Chaos: Strange attractors

- What is chaos?
- The Lorenz system.
- Strange attractors.

### 4 International dimension*

### 5. Teaching method and planned learning activities

Personal work

### 6. Assessment method and criteria

Individual assignements ("Homeworks")

Examination

Continuous assessment

### 7. Study material *

#### 7.1 Required reading

There exist lecture notes from last year that will be revised and weekly updated during this semester. Both versions (old and new) are available on the professor's WEBPAGE (not on blackboard!).

The homework sheets are available on the assistent's webpage.

**7.2 Optional reading**

The following study material can be studied voluntarily :The lecture notes contain references indicating further interesting/usefull books.

### 8. Contact information *

Sonja Hohloch,

email: sonja.hohloch AT uantwerpen.be