Dinâmica não Linear e Caos

Code:

MEM169

Acronym:

DNLC

Instance: 2003/2004 - 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	1	Plano de CPGEM 2002	1	1,5	5	-
MEM	6	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.

Main Bibliography

1. Nayfeh, A. H. and Balachandram, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: John Wiley and Sons, 1995.

2. Wiggins, Stephen; Introduction to applied nonlinear dynamical systems and chaos, Springer-Verlag.

Complementary Bibliography

3. Bergé, Pierre; Yves, Pomeau; Vidal, Christian. L'ordre dans le chaos: vers une approche déterministe de la turbulence. Paris: Hermann, 1988.

4. Nayfeh, A. H. and Mook. Nonlinear Oscillations. New York: John Wiley and Sons, 1995.

5. Seydel, R. Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos. Springer-Verlag, 1994.

6. Moon, F. C. Chaotic and Fractal Dynamics. An Introduction for Applied Scientists and Engineers. New York: John Wiley & Sons, 1992.

7. Guckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, 1997.

8. Szemplinska-Stupnicka, W. The Behaviour of Non-linear Vibrating Systems. Dordretch: Kluwer Academic Publishers, 1990. 2 volumes.

9. Nayfeh, A.H. Perturbation Methods. John Wiley and Sons.

Teaching methods and learning activities

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

Evaluation Type

Distributed evaluation with final exam

Calculation formula of final grade

0.4*Exam + 0.6*works