|Teoria||5||II semestre||Giacomo Canevari|
|Esercitazioni||1||II semestre||Giacomo Canevari|
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.
Module 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. Explicit 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 bifurcation in R2.
Linear systems in Rn, stable, unstable and central subspaces. Linearisation 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. scaling of vector fields and time reparametrisations.
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.
Module 8. Bifurcations and applications.
Definition of bifurcation, bifurcation at equilibria. Applications and numerical simulations.
Module 9. Calculus of variations in dimensione 1.
Module 10. Hamiltonian dynamics.
Hamiltonian systems, basic properties, Poisson bracket and canonical transformations. Lie conditions, generating functions, action-angle variables, integrability and Hamilton-Jacobi equation.
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 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.