Aims and scope

Discrete mathematical modeling plays a relevant role in many research fields.

The course aims to show how the mathematical analysis of discrete-time recursive laws allows substantial  foresight of qualitative and quantitative behavior of the evolution, together with the evaluation of sensitivity to initial conditions and parameters. In this perspective several tools for the analysis of discrete dynamical systems are introduced and they are used in the study of various models arising in applied sciences.

PROGRAM

1 – RECURSIVE PHENOMENA AND DIFFERENCE EQUATIONS

      Examples and motivations. Graphical analysis. Linear difference equations. Multi-step equations. Z transform.

2 – LINEAR SYSTEMS AND DISCRETE TRANSFORMS

      Derangements. Discrete Fourier Transform. Continued fractions, Euclidean algorithm, calendars. One-step nonlinear equations that can be reduced to the linear case.

3 – DISCRETE DYNAMICAL SYSTEMS

      Monotonicity and asymptotic analysis. Contraction mapping theorem. Phase diagram. Stability criteria based on derivatives. Hunting strategies. Periodic orbits.

      Explicit formulae for some nonlinear discrete dynamical systems.

4 – VECTOR-VALUED DISCRETE DYNAMICAL SYSTEMS

      Linear homogeneous systems and affine systems. Stability. Strictly positive matrices. Frobenius–Perron Theorem. Applications to genetics. Applications to demography.

5 – MARKOV CHAINS

      Stochastic matrices. Absorbing states. Invariant probability distributions. Markov-Kakutani Theorem. Asymptotic analysis. Irreducible matrices. Graphs. Adjacency matrix. Applications to network analysis.

6 – NONLINEAR DYNAMICAL SYSTEMS 

      Dynamics of logistic growth: h_a(x) = a (x-x²) . Sharkovsky Theorem. Bifurcations. Period doubling in logistic dynamics. Fatou Theorem. Stability of periodic orbits in logistic dynamics. 

7 – NONLINEAR DISCRETE DYNAMICAL SYSTEMS 

      Hyperbolic equilibria. Attractors. Topological conjugacy. Sensitivity to initial conditions. Topological mixing. Density of periodic orbits. Chaotic dynamics. Fatou Theorem. Iterations of a prescribed rotation

      on the circle. Jacobi Theorem. Doubling map. Dynamics of tent map. Logistic dynamics with parameter 4.

8 – LOGISTIC DYNAMICS WITH PARAMETER a>4

      Metric space of symbols. Symbolic dynamics. Shift map. Density of periodic orbits. Topological conjugacy of  when  a > 2 + \sqrt 5 .

9 - DISCRETE DYNAMICAL SYSTEMS IN THE COMPLEX PLANE

      Newton-Raphson method. Dynamical systems in the complex plane. Attraction basin for n-th complex roots of unity. Julia sets.

10 – FRACTAL DIMENSION AND SEMINARS

      Hausdorff dimension. Box counting dimension. Cantor-like sets. Hausdorff  distance between two sets. Kuratowki convergence of sets. Hutchinson self-similar-fractals. Hausdorff dimension  of self-similar fractals.

      Graphic generation of fractals through iteration of contraction mappings.

Preconditions

First level courses concerning Mathematical Analysis and Geometry. 

Exam procedure The final exam consists either in

an oral exposition about the whole program, with a closer examination on  two chapters to be chosen among the ones listed in the program (one of 6 in the first part and one of 4 in the second part),

or

an oral exposition about the whole program, with a closer examination on a chapter to be chosen among the ones listed in the program and on a different topic (connected to a chapter in the complementary part to the one containing the selected chapter), the topic must be agreed with the teacher before the end of the course (this part can be either a presentation during classes or a short dissertation or a project, delivered at least one week before the oral examination).