Dinâmica não Linear e Caos

Code:

MEM169

Acronym:

DNLC

Instance: 2004/2005 - 2S

Active?	Yes
Responsible unit:	Applied Mechanics Section
Course/CS Responsible:	Master in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
CPGEM	0	Plano de CPGEM 2002	1	1,5	5	-
MEM	0	Plano oficial a partir de 2002	1	1,5	5	-

Teaching language

Portuguese

Objectives

Teach fundamental aspects of the theory of non-linear dynamical systems, as well as numerical and experimental methodologies for their analysis. The theory has applications in a wide number of fields, but emphasis will be given to mechanical systems.

Program

1. Introduction. Non-linear dynamical systems.

2. Fundamental concepts of geometric theory and stability.
2.1 Autonomous and non-autonomous systems. 2.2 Stability of Lyapunov, assymptotic stability, stability of Poincaré. 2.3 Equilibrium points: centres, nodes, focus and saddle points. 2.4 Lymit cycles.

3. Methods of resolution of the equations of motion.
3.1 Perturbation method - asymptotic expansions. 3.2 Multiple Scales Method. 3.3 Harmonic balance method. 3.4 Numerical integration in the time domain. 3.5 Shooting method.

4. Periodic motions and methods to characterize motions.
4.1 Definition. Time history. 4.2 Phase plane. 4.3 Fourier Spectrum. 4.4 Poincaré Map. 4.5 Floquet theory. 4.6 Bifurcations of periodic solutions.

5. Quasi-periodic motions
5.1 Definition. 5.2 Time history; phase plane; Fourier spectrum and Poincaré map.

6. Chaos.
6.1 Definition. Routes to chaos. 6.2 Time history; Phase plane; Fourier Spectrum and Poincaré Maps. 6.3 Lyapunov exponents.

Mandatory literature

Nayfeh, Ali Hasan; Applied nonlinear dynamics. ISBN: 0-471-59348-6
Thomsen, Jon Juel; Vibrations and stability. ISBN: 3-540-40140-7

Complementary Bibliography

Meirovitch, Leonard; Elements of vibration analysis. ISBN: 0-07-041342-8
Nayfeh, A. H. and Mook; Nonlinear Oscillations
Wiggins, Stephen; Introduction to applied nonlinear dynamical systems and chaos. ISBN: 0387-00177-8
Bathe, Klaus-Jurgen; Finite element procedures. ISBN: 0-13-301458-4
Bergé, Pierre; Yves, Pomeau; Vidal, Christian; L'ordre dans le chaos: vers une approche déterministe de la turbulence.
Medio, Alfredo; Non-Linear Dynamics, Cambridge University Press, 2001

Teaching methods and learning activities

Exposition of theory, with some demonstrative examples. Computational applicatons will be carried out and suggested to the students.

Software

Os trabalhos práticos serão efectuados de preferência com recurso ao Maple ou ao Matlab. No entanto também se aceita o uso do Fortran, C, ou outra linguagem à escolha do aluno.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Subject Classes	Participação presencial	24,00
	Total:	-	0,00

Calculation formula of final grade

0.4*Exam + 0.6*works