# Dynamical Systems

**Academic Year 2023/2024**- Teacher:

**Vittorio ROMANO**

## Expected Learning Outcomes

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.

## Course Structure

Mainly frontal lectures. Simple study cases will be tackled in a MATLAB environment.

Theory lessons: 35 hours.

Exercises: 12 hours.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Learning assessment may also be carried out on line, should the conditions require it.

IMPORTANT: in order to guarantee equal opportunities to the students with handicap and/or any form of disability, such students may ask to talk to the teacher to program suitable actions. The interested students may also contact prof. Filippo Stanco.

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

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 recurrence theorem of Poincaré. Limit sets and attractors.

Limit orbit. Periodic orbits, limit cycles, separators.

Stablty of limit cycles. Homoclinic and heteroclinic orbits. 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.

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

## Textbook Information

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

## Course Planning

Subjects | Text References | |
---|---|---|

1 | 1. Introduction to dynamical systems. Definitions and examples: bank account,, armonic oscillator, simple pendulum, population dynamics (Malthus model and logistic equations) , dynamics of couple of populations (prey-predator model). | 1 |

2 | Linear dynamical systems. One dimensional systems: discrete case, analytical method and graphic analysis; continuous case. Dynamical systems in dimension greater than one. Exponential matrix. Linear stability and instability. geometrical representation as vectorial fields. Phase portrait. Saddle, node, focuss, center, well and source points. | 1,2,3 |

3 | Nonlinear systems. Fixed points: discrete and continuous case. Stability and instability of the critical points. Linearization. Hyperbolic equilibrium points. Statement of the theorem of the stable manifold and of the theorem of di Hartman-Grobman. Topogically equivalent and conjugate dynamical systems. Examples. Esempi. Method of Lyapunov. Theorems of Lyapunov regarding the stability, instability and asymptotic stability of critical points. Dirichlet-Lagrange theorem. Sistems of gradient type and Hamiltonian type. Alfa-limit and omega-limit sets of a trajectory of a dynamical system. | 1,2,3 |

4 | Structurally style and unstable dynamical systems. Bifurcations. Parametro di biforcazione. Diagram of bifurcation. saddle-node, transcritical, pitchfork and Hopf bifurcations. Statement of the Hopf bifurcation theorem. Lorenz system, strange attractor and caos. | 1,2 |

5 | Fractals. Cantor's set of the third medium. Fractal dimension. Triangle of Sierpinski. Snowflake of Koch. Contractions. Theorem of Banach-Caccioppoli. Compact sets and distance of Hausdorff. Systems of iterate functions. Applications. | 1 |