To show the organization of the course that includes this module, follow this link Course organization
The aim of the first module is to deepen the knowledge and skills especially in the modern theory of dynamical systems and give the student a solid appreciation of the deep connections between mathematics and other scientific disciplines, both in terms of the mathematical problems that they inspire and the important role that mathematics plays in scientific research and industry. Mathematical software tools, and others, will be used to implement algorithms for the solution of the real world problems studied during the course. At the end of the course the student is expected to be able to complete professional and technical tasks of a high level in the context of mathematical modelling and computation, both working alone and in groups. In particular the student will be able to write a model of a real problem, to recognise the effective parameters and analyse the model and its possible implications. The second module wants to provide sufficient theoretical and numerical background for the optimal control of dynamical systems. Such problems will be developed by means of real application examples, and recent research studies. At the end of the course students will be able to decide which numerical method is suitable for the solution of some specific optimal control problems. He/She will be able to provide theoretical results on the controllability and stability of certain optimal control problem and numerical methods. He/She will be able to develop his/her own code, and capable choose the appropriate optimization method for each application shown during the course.
1. Review of Dynamical systems: vector fields, flows, equilibria and their tability, periodic orbits, phase portrait and first integrals. Estimation on orbit separation. Translation of the circle. Bifurcation in 1 dimension . Numerical integrations and phase portraits in dimensions 2.
2. Biforcation and limit cycles: bifurcations in dimensions 2, limit cycles, Poincare`-Bendixon Theorem. Numerical aspects and applications.
3. Nonautonomous vector fields: extended phase space, flow box and Poincare' maps, quotient phase space. NUmerical integrations and applications.
4. Basic aspects on discrete dynamics: the standard map.
|Stephen Lynch||Dynamical Systems with Applications using Mathematica® (Edizione 1)||Birkhäuser||2017||978-3-319-87089-2||Access to the Notebook used in the book https://www.springer.com/gp/book/9783319614847|
|Stephen Lynch||Dynamical Systems with Applications using MATLAB® (Edizione 2)||Birkhäuser||2014||978-3-319-33041-9||Access to the Matlab files used in the book https://www.springer.com/gp/book/9783319068190|
|Stephen Lynch||Dynamical Systems with Applications using Python® (Edizione 1)||Birkhäuser||2018||978-3-030-08624-4||Access to the python files used in the book https://www.springer.com/gp/book/9783319781440|
|S. Strogatz||Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering||Westview Press||2010|
The student is expected to demonstrate the ability to mathematically formalize and solve models used in several scientific discipline, using, adapting and developing the models and advanced methods discussed during the lectures. To that end the final evaluation will consist in a written and oral exam.
Written exam: One question/exercise for each part of the course (Part I and Part II), the solution will possible require the use of computer.
Oral exam: Subject of student choice and discussion of the written exam with questions.
The subject of student choice can be substituted with the development of a small-project to be decided together with the teacher.