This course is of 8 CFU and is associated with a version of 5 CFU named 097681 - DISCRETE DYNAMICAL MODELS (5 CFU):

the current sheet specifies goals, program and expected learning results for both courses.

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.

The course has an innovative section of type "laboratorio progettuale", including elaboration of students projects and their presentation (flipped classroom), experts seminars.

For both versions of the course (8 CFU and 5 CFU), we expect that the students learn:

Ability to model recursive phenomena and deal with them either by graphical analysis or by analytical techniques.

Solving difference equations in the linear case and in some nonlinear case when by explicit formulae are available.

Applying properties of permutations and derangements to combinatorial problems.

Performing qualitative analysis of nonlinear scalar discrete dynamical systems by monotonicity and analytical techniques: phase diagrams, orbits, stability criteria, discrete transforms.

Exploiting Linear Algebra tools in the analysis of demographical models, networks and processes described by Markov chains with finite states.

Knowledge of basic properties of discrete logistic growth: equilibria, orbits, stability, bifurcations, topological conjugacy, symbolic dynamics, fractal sets.

Critical attitude in the choice of models and on their reliability: well-posedness and stability versus sensitive dependance on data.

Projects presentation with flipped classroom.

We report the topics discussed in both versions of the course (8 CFU and 5 CFU). We use '(*)' for the topics that will not be discussed in the 5 CFU version.

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 - 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.

11 (*)  Follow-up presentations by students and Seminars by experts in the field are planned too.

Preconditions

First level courses concerning Mathematical Analysis and Geometry. 

The final exam for the 8 CFU version 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 lab, delivered at least one week before the oral examination).

The student must show abilities in: modeling recursive phenomena; dealing with them either by graphical analysis or by solving difference equations; applying properties of permutations to combinatorial problems; performing qualitative analysis of nonlinear scalar discrete dynamical systems by monotonicity and differential techniques (phase diagram, orbits, stability criteria, discrete transforms); exploiting Linear Algebra tools in the analysis of demographical models, networks and processes described by Markov chains with finite states ; knowledge of basic properties of discrete logistic growth (equilibria, orbits, stability, bifurcations, topological conjugacy, symbolic dynamics, discrete dynamical systems in the complex plane, dimension of self-similar fractal sets); acquisition of a critical attitude in the choice of models and on their reliability (well-posedness and stability versus sensitive dependance on data).

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The final exam for the 5 CFU version consists in an oral exposition about all the chapters listed without (*) in the program, with a closer examination on one chapter to be chosen among the ones listed without (*) in the program.

The student must show abilities in: modeling recursive phenomena; dealing with them either by graphical analysis or by solving difference equations; applying properties of permutations to combinatorial problems; performing qualitative analysis of nonlinear scalar discrete dynamical systems by monotonicity and differential techniques (phase diagram, orbits, stability criteria, discrete transforms); exploiting Linear Algebra tools in the analysis of demographical models, networks and processes described by Markov chains with finite states ; knowledge of basic properties of discrete logistic growth (equilibria, orbits, stability, bifurcations, topological conjugacy, symbolic dynamics); acquisition of a critical attitude in the choice of models and on their reliability (well-posedness and stability versus sensitive dependance on data).