Dynamical Systems

Code:

EMG0038

Acronym:

SD

Keywords
Classification	Keyword
OFICIAL	Physical Sciences (Physics)
OFICIAL	Physical Sciences (Mathematics)

Instance: 2023/2024 - 2S

Active?	Yes
Responsible unit:	Mining Engineering Department
Course/CS Responsible:	Bachelor in Mining and Geo-Environmental Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EMG	19	Plano de estudos oficial a partir de 2008/09	2	-	4,5	39	121,5

Teaching language

Suitable for English-speaking students
Obs.: Aulas em inglês sempre que se verifique a presença de estudantes não falantes da lingua portuguesa.

Objectives

To recall the concepts of function, independent variable, dependent variable
 To learn the concept of dynamical system
Classify the dynamical systems from the ODE and the initial conditions
 To recall the main analytical methods of solving 1s order ODE
 Analytically solve linear DEs of higher order than the first one
To learn how to operate with discrete dynamical systems and how to solve differences equations
To operate with integral transformations
To review the meaning of ODE
To identify order of a dynamical system
To distinguish between linear nonlinear, continuous discrete systems
To describe the methods applicable to the resolution of each DE and the differences
Mathematically translate a 1st order kinetic process
To interpret physically the solution of a DE and a difference equation
To use CAS in the implementation and resolution of DEs
To operate with complex abstract entities such as integral transformations

Learning outcomes and competences

Understanding the concept of dynamical system; classifying dynamical systems from the differential equation and initial conditions; operating with integral transformations; distinguishing linear from non-linear, continuous from discrete systems; translating a 1st order kinetic process mathematically; interpreting the solution of a differential equation in physical terms; using algebraic manipulators to implement and solve differential equations.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Know how to operate with derivatives and integrals.

Program

- First order ordinary differential equations: their formulation, interpretation and resolution.

- Differential equations of order n (linear of variable coefficients and linear of constant coefficients). Physical example of a 2nd order differential equation (system of free oscillations). Systems of linear differential equations of constant coefficients.

- Physical interpretation of differential equations.

- Frequency response of continuous time systems. Heaviside Step, Pedestal, and Dirac Impulse functions. Convolution of continuous temporal systems – step response and impulse response.

- Laplace transform. Application of the Laplace transform to the resolution of differential equations.

- Representation of discrete systems. Linear differences equations. Linear 1st order differences equations. Examples of 1st order linear discrete systems. Linear 2nd order differences equations.

Mandatory literature

António Fiúza; Dinâmica de Sistemas, 2000

Complementary Bibliography

Shepley L. Ross; Introduction to ordinary differential equations. ISBN: 0-471-09881-7
Jaime Enrique Villate Matiz; Introdução aos sistemas dinâmicos. ISBN: 972-99396-0-8
Hirsch, M. W. , Smale, S., & Devaney, R. L. ; Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier Academic Press., 2004
Bronson, Richard; Costa, Gabriel; Equações Diferenciais, Bookman, 2015. ISBN: 978-8577801831

Teaching methods and learning activities

Theoretical-practical classes will be distributed weekly over two periods on different days. The first part of each TP class will be used for oral presentation of the contents to students, the second part will be occupied with applications of the subjects taught, with emphasis on the diversification of implementation methods - use of computers (algebraic manipulators), solving exercises with a volunteer (student or teacher) working on the board, and the remaining students working in place under the supervision and monitoring of the teacher. Exercise statements are previously available on online platforms (moodle and sigarra).

Software

WxMaxima

keywords

Technological sciences > Engineering > Systems engineering > Systems theory

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	10,00
Teste	90,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	82,50
Frequência das aulas	39,00
Total:	121,50

Eligibility for exams

To obtain attence, the student cannot exceed the limit number of absences - 25% of the number of classes given.

Calculation formula of final grade

The first call grade will be the weighted average of the distributed evaluation components.
N1 = 0.4 * T1 + 0.5 * T2 + 0.1 * P
Being T1 and T2 mandatory tests; P stands for class participation and homework.
Students whose 1st call grade (N1) is less than 10 will be subjected to a second call exam.

Examinations or Special Assignments

Not foreseen.

Internship work/project

Not applicable.

Special assessment (TE, DA, ...)

Students with special status, in addition to the evaluation required of regular students, can take a global exam on all the subjects taught during a special exam period.

Classification improvement

In a final exam covering all the subjects taught.
Isolated components of the distributed assessment are not subject to improvement of classification.