PHYSICS OF DYNAMICAL SYSTEMS
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
Basic knowledge of Calculus, Classical mechanics and Analytical mechanics (especially Hamiltonian dynamics).
A oral exam, where the student must discuss the course topics in depth and quantitatively, underlining their most significant aspects.
Before sustaining the oral exam, students are also required to work at a small numerical project assigned by instructor. They will prepare a short report that will be discussed at the beginning of the oral exam. They can chose to work at this project and prepare the individually or in groups of two (but the consequent discussion is always individual). Successful projects will demonstrate familiarity with the numerical techniques discussed in the course and the ability to successfully analyse the proposed dynamical system/problem.
To successfully pass the exam students should successfully complete the numerical project and know all the topics presented in the course. The deeper the knowledge the better the evaluation. Full mark with laude is assigned only to students that accomplish completely the learning outcomes discussed above, showing a solid qualitative and quantitative comprehension of the course topics and the ability to re-elaborate them.
The theory of dynamical systems represents a fundamental background for the study and understanding of a wide range of physical, biological and chemical phenomena. It also bears noteworthy applications to social and economic sciences.
At a more fundamental level, the discovery of deterministic chaos, with its strong sensitivity to initial conditions, shows that not all deterministic dynamical systems are predictable. This notion is an essential part in the modern understanding of the world around us.
This course aims to introduce the fundamental notions necessary for the qualitative and quantitative understanding of the physics of dynamical systems, from regular to chaotic motions. Numerical examples and algorithms for the study of nonlinear dynamics will also be introduced.
On completion of the course the student should be able to
• Recognize and classify the basic classes of regular and chaotic motion.
• Show a qualitative and quantitative understanding of chaotic dynamics
• Apply the basic concepts and methods that underlie the study of dynamical system, such as: linear stability theory, bifurcation theory, ergodic theory, fractals, characteristic Lyapunov exponents, chaos in Hamiltonian systems and the transition to chaos.
• Analyze numerically simple models of chaotic dynamics.
1. Introduction to dynamical systems [14 h]:
Dynamics in phase space, simple attractors, linear stability analysis, Poincarè map. Introduction to numerical simulations. Elements of bifurcation theory, normal forms.
2. Introduction to chaos: [8 h]
Logistic map, Lorenz attractor, Henon map. Numerical simulations.
3. Characterization of chaos [16 h]:
Fractal attractors, probability measure, ergodicity, mixing, characteristic Lyapunov exponents.
4. Hamiltonian dynamics [6h]:
Symplectic structure and canonic transformations. Poincare’ recurrence theorem. Elements of KAM theory. Symplectic algorithms.
5. Advanced topics [4 h].
Elements of time-series analysis: embedding techniques. Transition to chaos: period doubling, intermittency.
M.Cencini, F. Cecconi and A. Vulpiani CHAOS: From Simple Models to Complex Systems (World Scientific, Singapore 2009)
Steven H. Strogatz: Nonlinear dynamics and chaos (Taylor & Francis, 2016)
E. Ott. Chaos in dynamical systems (Cambridge University Press, 2002).
The course is essentially based on lectures, during which the teacher
presents the contents of the course in full detail, including mathematical
derivations
For questions/discussion/comments, students are invited to contact the teacher via email at the following address: francesco.ginelli@uninsubria.it
You are also welcomed to directly visit me in my office, but to be sure to find me is always better to contact me by email beforehand.