Dynamical Systems (2018/2019)



Course code
4S00244
Credits
9
Coordinator
Nicola Sansonetto
Other available courses
Other available courses
    Academic sector
    MAT/05 - MATHEMATICAL ANALYSIS
    Language of instruction
    Italian
    Teaching is organised as follows:
    Activity Credits Period Academic staff Timetable
    Parte I 6 II semestre Marta Zoppello

    Go to lesson schedule

    Esercitazioni parte II 1 II semestre Nicola Sansonetto

    Go to lesson schedule

    Esercitazioni 2 parte II 2 II semestre Giacomo Canevari

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    Learning outcomes

    The aim of the course is the introduction of the theory and of some applications of continuous and discrete dynamical
    systems, that describe the time evolution of quantitative variables. At the end of the course a student will be able to
    investigate the stability and the character of an equilibrium and to produce and investigate the qualitative analysis of a
    system of ordinary differential equations and the phase portrait of a dynamical system in dimension 1 and 2.
    Moreover a student will be able to study the presence and the nature of limit cycles and to analyse some basic
    applications of dynamical systems arising from population dynamics, mechanics and traffic flows.
    Eventually a student will be also able to produce proofs using the typical tools of modern dynamical

    Syllabus

    1. Basic principles.
    The Cauchy Problem. Completeness. Flows and orbits. Re-parametrization. Local rectifiability Theorem. First examples: exponential growth, the logistic equation, the Lotka–Volterra equation, the SIS and SIR models, car-following…
    2. Models and examples
    One-dimensional systems. Conservative systems with one degree of freedom, Newtonian systems. Linear systems: dimension 1, 2, n. Non-linear systems in R^2.
    3. Discrete-time systems.
    Definitions. Examples: bacterial growth, Fibonacci, structured populations, AIMD…
    Linear systems and z-transform. Stability.
    4. Stability
    Definition. Lyapunov theory. Alpha and Omega-limits. Poincaré-Bendixson Theorem.

    Assessment methods and criteria

    A written exam with exercises: phase portrait in 2D for a non-linear dynamical system; computation of trajectories and stability for a discrete-time system, phase portrait in 2D for a non-linear dynamical system; computation of trajectories and stability for a discrete-time system; stability analysis for a system.
    The written exam tests the following learning outcomes:
    - To have adequate analytical skills;
    - To have adequate computational skills;
    - To be able to translate problems from natural language to mathematical formulations;
    - To be able to define and develop mathematical models for physics and natural sciences.

    An oral exam with 2-3 theoretical questions. The oral exam is compulsory and must be completed within the session
    in which the written part has been done.
    The oral exam tests the following learning outcomes:
    - To be able to present precise proofs and recognise them.

    Reference books
    Activity Author Title Publisher Year ISBN Note
    Parte I M.W. Hirsch e S. Smale Differential equations, dynamical systems, and linear algebra Academic Press 1974
    Parte I S. Strogatz Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Westview Press 2010
    Parte I F. Fasso` Primo sguardo ai sistemi dinamici CLEUP 2016
    Esercitazioni parte II M.W. Hirsch e S. Smale Differential equations, dynamical systems, and linear algebra Academic Press 1974
    Esercitazioni parte II S. Strogatz Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Westview Press 2010
    Esercitazioni parte II F. Fasso` Primo sguardo ai sistemi dinamici CLEUP 2016
    Esercitazioni 2 parte II M.W. Hirsch e S. Smale Differential equations, dynamical systems, and linear algebra Academic Press 1974
    Esercitazioni 2 parte II S. Strogatz Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Westview Press 2010
    Esercitazioni 2 parte II F. Fasso` Primo sguardo ai sistemi dinamici CLEUP 2016

    STUDENT MODULE EVALUATION - 2017/2018