Study location	Germany, Berlin
Type	On Campus, full-time
Nominal duration	4 weeks
Study language	English
Awards	Summer School
Accreditation	6 ECTS

Tuition fee	€2,150 per programme The program price consists of the course/tuition fee (student or working professional, see details below) plus the registration fee (€60). Student course/tuition fee: €2090 Working professional course/tuition fee: €2510 This course/tuition fee covers the course, course materials and a cultural program.
Registration fee	€60 one-time The registration fee is in addition to the course/tuition fee and covers the processing of your application. It is payable upon registration. Please note that the registration fee is non-refundable.

Entry qualification	At least one year of university experience or equivalent work experience The entry qualification documents are accepted in the following languages: English / German. Please upload one of the following documents: University degree Transcript of records University enrollment certificate Upload copies in a word or pdf format

Language requirements

English All applicants are required to upload a document or certificate to demonstrate their proficiency in English language. If you are a non-native English speaker, you must prove you have a score equivalent to the level B2 or above in the European system (the Common European Framework of Reference for Languages, or CEFR), or provide evidence that you’ve undertaken an equivalent degree/studies in English. A list of scores from the main providers is included for reference. Certificates from other providers are also accepted. CEFR: B2 IELTS: 5-6 Cambridge exam: First certificate in English FCE (A-C) TOEFL iBT: 87 TOEFL Paper based: 600 Chinese CET – 4: 493 Chinese CET – 6: 450 TOEIC: 685 More details: www.tu-berlin.de/menue/summer_university/requirements/ If you are a native English speaker, please select this during registration. You will then be exempt from having to upload proof of English level.

Other requirements

Additional prerequisites here: Knowledge of calculus (drawing of functions, differentiation and basic integration) is needed; An understanding of linear algebra (eigenvalues, eigenvectors) is recommended. Some basic concepts will be recalled in the first lectures; A basic knowledge of solution methods for linear, ordinary differential equations (ODEs) is also helpful but not strictly needed. These methods cannot generally be used for solving nonlinear ODEs, and this course will teach other, less canonical methods for their solution using geometrical intuition.

More information	tu-berlin.de/menue/summer_university

Overview

Nonlinear dynamics governs the evolution of a tremendously large number of phenomena in our world. These phenomena are observed in fluid dynamics, biology, mechanical and acoustic vibrations, finance, and many more areas. The beauty of nonlinear dynamics is that it is universal: even if developed or learned in a specific context, the same theory and tools can be then used to understand the nonlinear effects of a completely different, seemingly unrelated subject. The course aims to (i) explain the foundations of nonlinear dynamical systems in relation to the solution of nonlinear differential equations describing physical systems of relevance for engineering applications, and to (ii) introduce some of the graphical, mathematical and numerical methods used to study and model nonlinear phenomena. Geometric intuition will aid in keeping the mathematical formalism to a minimum. Real-world examples borrowed from state-of-the-art research in various fields, with a focus on engineering, will aid in attracting the student’s interest and in demonstrating the usefulness of the thought methods. In this sense, the “teaching by example” approach will be adopted.

Learning goals:

• Learn the fundamentals of nonlinear dynamics, and to identify fixed point, periodic and non-periodic solutions.
• Be able to discuss both qualitatively and quantitatively the nature and solutions of nonlinear dynamical systems.
• Learn how graphical methods can aid in the solution of these systems

Programme structure

The key topics covered in the course are:

1. Importance of nonlinear effects in physical phenomena
2. Classification of fixed points and bifurcations
3. Understanding of bi-stability and triggering
4. Drawing and interpreting phase portraits of low-dimensional dynamical systems
5. Qualitative understanding of the method of averaging
6. Periodic and non-periodic solutions
7. Chaotic systems.

The teaching will combine frontal lectures to introduce new topics and applied lectures in which the theory is applied to problems of practical interest in engineering. The class will also carry out exercises and small programming tasks in groups during the course. In this sense, a “teaching by examples” approach will be followed. A final project will be carried out in groups and presented to the rest of the class.