# Dynamical Systems (2019/2020)

Course code
4S00244
Credits
9
Coordinator
Nicola Sansonetto
Other available courses
Other available courses
MAT/05 - MATHEMATICAL ANALYSIS
Language of instruction
Italian
Teaching is organised as follows:
Activity Credits Period Academic staff Timetable
Parte I esercitazioni 1 II semestre Giacomo Canevari

Parte I teoria 5 II semestre Nicola Sansonetto

Parte II teoria 1 II semestre Nicola Sansonetto

Parte II Esercitazioni 2 II semestre Ancora Da Definire

### 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 systems and will be able to read and report specific books and articles on dynamical systems and related applications.

### Syllabus

Part 1
Modul2 1. Complements of ordinary differential equations.
First and second order differential equations. Methods of the variations of the constants. Existence and uniqueness theorem. Qualitative analysis of ODE: maximal solutions, Gronwall’s Lemma. Esplicit solutions of particular equations: separations of variable, Riccati and total equations. Linear systems.

Module 2. Vector fields and ODE.
Orbits and phase space. Equilibria, phase portrait in 1 dimension. ODE of the second order and their equilibria. LInearisation about an equilibrium and periodic solutions of an ODE.

Module 3. Linear systems.
Linear systems in in R2, real and complex eigenvalues. Elements of Jordan theory. Diagram of biforcation in R2.
Linear systems in Rn, stable, unstable and central subspeces. Linearization about an equilibrium.

Module 4. Flows and flows conjugations.
Flow of a vector field. Dependance on the parameters. time dependent vector fields.
Change of coordinates, conjugations of flows, pull-back and push-forward of functions and vector fields. Time dependent change of coordinates.riscalamenti di campi vettoriali e riparametrizzazioni del tempo.
Rectification theorem.

Module 5. First integrals.
Invariant sets, first integrals and Lie derivative. Invariant foliations, reduction of order. First integrals and attractive equilibria.

Module 6. 1-dimensional Newton equation. Phase portrait in the conservative case. Linearisation. Reduction of order. Systems with friction.

Module 7. Stability theory.
Lyapunov Stability, Lyapunov functions and spectral method.

Part 2.
Module 8. Calculus of variations.

Module 9. Bifurcations and applications.
Definition of bifurcation, bifurcation at equilibria. Applications.

Module 10. 2-dimensional flows.
Linear flow on T2. Population dynamics and Lotka-Volterra system. Critical point in R2 and bifurcations
Limit cycle: Lienard and Van der Pol equation, Hopf bifurcation. alpha and omega sets, Poincare ́–Bendixon Theorem

Module11. Hamiltonian dynamics.
Hamiltonian systems, basic properties, Poisson bracket and canonical transformations. Lie conditions, generating functions, action-angle variables, integrability and Hamilton-Jacobi equation.

### 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 vector field, phase portrait in 2D for a non-linear dynamical system. Stability analysis for a system. Linearisation about equilibria and analysis of the linearised system.
Change of coordinates, first integrals, reduction of order. Hamiltonian and Lagrangian dynamics.
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.

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