PHYSICS OF DYNAMICAL SYSTEMS
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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.
We expect students to develop an in-depth understanding of the topics covered, together with the ability to deal individually with non-linear dynamics problems through the correct use of the technical tools introduced.
Basic knowledge of Calculus, Classical mechanics and Analytical mechanics (especially Hamiltonian dynamics).
Introduction to dynamical systems: Dynamics in phase space, simple attractors, linear stability, Poincarè map.
Elements of bifurcation theory, normal forms.
Introduction to chaos: logistic map, Lorenz attractor, numerical approaches.
Characterization of chaos: fractal attractors, probability measure, ergodicity, mixing, Lyapunov exponents.
Transition to chaos and deterministic systems, period doubling.
Hamiltonian dynamics. Canonical transformations. Action-angle variables. Integrable systems. Elements of KAM theory.
Introduction to space-time chaos.
Course topics will be discussed in lectures, supplemented by home exercises, which will be assigned periodically to the students.
A oral exam, where the student must discuss the proposed topics in depth and quantitatively, underlining their most significant aspects.
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).
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