DYNAMICAL SYSTEMS

Fornire gli strumenti affinchè gli studenti possano formulare modelli per semplici problemi tratti dalle scienze applicate tramitei sistemi dinamici, discreti o continui, finito dimensionali. In particolare si studieranno sistemi lineari e non lineari. Si studierà la stabilità dei punti di equilibrio, l'esistenza di attrattori strani e gli insiemi frattali.

The course aims to provide the basic knowledge for formulating, for simple problems arising from the applied sciences, models repesented by finie dimensional (discrete or continuos) dynamical systems. In particular, linear and nonlonear systems will be analyzed. The stability of the equilbrium points, the existence of strange attractors and fractal systems will be studied.

In particular, the course aims to allow the student to acquire the following skills:

knowledge and understanding: knowledge of results and fundamental methods. Skill of understanding problems and to extract the major features. Skill of reading, undertanding and analyzing a subject in the related literature and present it in a clear and accurate way.

applying knowledge and understanding: skill of elaborainge new example or solving novel theoretical exsercise, looking for the most appopriate methods and applying them in an appropriate way.

making judgements: To be able of devise proposals suited to correctly interprete complex problems in the framework of dynamical systems and their applications. To be able to formulate autonomously adequate judgements on the applicablity of mathematical models to theoretical or real situations.

communication skills: skills of presenting arguments, problems, ideas and solutions in mathematical terms with clarity and accuracy and with procedures suited for the audience, both in an oral and a written form. Skill of clearly motivating the choice of the strategy, method and contents, along with the employed computational tools.

learning skills: reading and analyzing a subject in the engineering literature involving applied mathematics. To tackle in an autonomuous way the systematic study of arguments not previously treated. To acquire a degree of autonomy such that the student can be able to start with an autonomuos reserach activity.

Examples of models based on finite dimensional (discrete or continuous) dynamical systems.

Linear and nonlinear dynamical systems.

Equilibria and their stability.

Periodicity and caos.

Fractals.

1. Examples of dynamical systems:
bank account, harmonic oscillator, simple pendulum, economic growth, population dynamics.

2. Linear dynamical systems. Associated matrix and its eigenvalues. Jordan's canonical forms. Matrix esponential. Linear stability and instability. Representation of the solutions as vectorial fields. Phase portrait. Saddles, nodes, focis, centers, wells, springs.

3. Nonlinear dynamica systems. Fixed points and their stability/instability. Linearization around a hyperbolic equilibrium point. Theorems of the stable manifold and of
Hartman-Grobman. Dynamical systems topologycally equivalent and conjugate. Examples. Lyapunov's method. Theorems of Lyapunov.
Theorems of Dirichlet-Lagrange. Systems of gradien type. Alpha-limit and omega-limit sets of a trajectory. Hamiltonian systems and applications. Limit sets and attractors.
Limit orbit. Periodic orbits, limit cycles, separators.
Stablty of limit cycles. Homoclinic and heteroclinic orbits. Poincaré's map. Theorem on the existence of Poincaré's map and theorem of Poincaré-Bendixon. Mathematical models of pandemics.

4.Nonlinear systems, periodicity and caos. Bifurcation parameters.
Diagram of bifurcation. Saddle-node, trancritical, picthfork and Hopf bifurcation. Hopf bifurcation theorem. Lorenz system and strange attractor. Discrete case.
Tangent, pichtfork and transitical bifucations. Caos and synbolic dynamics.

5. Fractals. Cantor's set. Triangle of Sierpinski.
Koch snowflake. Contraction theorem of Banach-Caccioppoli. Spectral norm.
Compact sets and Hausdorff metrics. Iterate function systems.
Kolmogorov fractal dimension. Applications.

1. E. Scheinerman, Invitation to Dynamical Systems, available online: http://www.ams.jhu.edu/∼ers/invite/book.pdf

2. L. Perko, Differential equations and dynamical systems, 3rd ed. - New York: Springer-Verlag, 2001.

3. F. Ganthmaker, Lectures in Analytical Mechanics, MIR 1975