Teaching code

4S00244

Credits

9

Coordinator

Giacomo Canevari

Language

Italian

Also offered in courses:

• Dynamical Systems of the course Bachelor's degree in Applied Mathematics

Scientific Disciplinary Sector (SSD)

MAT/05 - MATHEMATICAL ANALYSIS

Courses Single

Authorized

Moodle Seminars 0

The teaching is organized as follows:

Learning objectives

The aim of the course is to introduce the theory and some applications of dynamical systems, which describe the time evolution of quantitative variables. By the end of the course, the students will be able to investigate the stability and the character of an equilibrium, the qualitative analysis of a system of ordinary differential equations, the phase portrait of a (parametric) dynamical system in dimension 1 and 2, and to analyse finite-dimensional Hamiltonian systems. Moreover, the students will be able to study some basic applications of dynamical systems arising from population dynamics, mechanics and traffic flows. Finally, students 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.

Prerequisites and basic notions

The material covered in first-year and second-year, first-semester courses - in particular, Mathematical Analysis 1 and 2, Linear Algebra.

Program

Part I
1. Topics in the theory of ordinary differential equations
Qualitative analysis of ODE: existence and uniqueness of solutions; maximal and global solutions; Gronwall’s Lemma; continuous dependence on the initial data.
2. Vector fields and ordinary differential equations
Vector fields: phase space, integral curves, orbits, equilibria, phase portrait. 1-dimensional examples of phase portraits. Second-order systems of differential equations; phase-space analysis and equilibria.
3. Linear systems
Linearisation of a vector field about an equilibrium. Classification of two-dimensional linear systems (over the real numbers) that are diagonalisable over the complex numbers. (If time permits, we will briefly discuss the nilpotent case as well.) n-dimensional linear systems: invariant subspace decomposition; the stable, unstable and central subspaces. Comparing a vector field with its linearisation about a hyperbolic equilibrium.
4. Flow of a vector field
Flow of a vector field. Change of coordinates: conjugate vector fields; pull-back and push-forward of a vector field by a diffeomorphism. Non-autonomous differential equations: time-dependent change of coordinates; scaling of vector fields and time reparametrisations. The local rectification theorem.
5. First integrals
Invariant sets; first integrals; Lie derivative. Invariant foliations; reduction of the order. First integrals and attractive equilibria.
6. Stability theory
Stability 'à la Lyapunov' of an equilibrium; the method of Lyapunov functions; the spectral method. Applications and examples.
7. 1-dimensional Newton equation.
Phase portraits of the 1-dimensional Newton equation, in the conservative case. Linearisation. Reduction of the order. Systems with friction.
Part II
8. Bifurcations
Bifurcatios from equilibria, with 1-dimensional examples; applications.
9. Introduction to the 1-dimensional Calculus of Variations
The indirect method for one-dimensional integral functionals. Necessary conditions for the existence of minimisers: the Euler-Lagrange equations. Jacobi integral; conservation laws. Geodesics on a surface.
10. Hamiltonian systems
Hamiltonian vector fields. Legendre transform. Poisson brackets. Canonical transformations. Lie conditions, generating functions. The Hamilton-Jacobi equations. Integrability. Geometry of the phase space: Liouville's theorem and Poincaré's recurrence theorem.

Bibliography

Didactic methods

Lectures and exercises (often, combined).

Learning assessment procedures

The exam consists of a written part, with exercises and/or questions on the course content, and an oral part.

Evaluation criteria

Knowledge of the content of the course. Ability to solve simple problems. Logical rigour. Clarity of presentation.

Criteria for the composition of the final grade

Please see the Italian version.

Exam language

Italiano