Marie Curie Europe

TC4: Advanced Nonlinear Dynamics and Chaotic Dynamical Systems

TC4: Advanced Nonlinear Dynamics and Chaotic Dynamical Systems

A training course coordinated by Claude-Henri Lamarque
time:   March 30 - April 3, 2009
location: ENTPE Lyon, France


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Aim of the course
This course will focus on advanced mathematical techniques for analyzing the nonlinear dynamics of maps and vector fields. The course will combine theoretical treatment of concepts and techniques, and applications in mechanics.
 
Topics will include: 
 
Asymptotic methods for analyzing dynamical systems; basics of the method of multiple scales and demonstration of the use of this method for studying bifurcations of free and forced single- and multi-degree-of-freedom DOF dynamical systems.
 
Theory of normal forms for vector fields and maps; co-dimension 1 bifurcations of vector fields and maps; saddle-node, transcritical, pitchfork, and Hopf bifurcations; period-doubling bifurcation for maps; introduction to co-dimension-2 local bifurcations; aspects of Global Bifurcations and Chaos, homoclinic bifurcations. 
 
Theory of nonlinear normal modes (NNMs), bifurcations, instability and localization of NNMs; the concept of targeted energy transfer; NNMs and soliton-like oscillations (breathers); Applications of the theory of NNMs in simple mechanical systems (coupled oscillatory chains with smooth potential, beams and arches. 
 
Applications of advanced mathematical and signal processing tools to practical problems in nonlinear dynamics;. effects of friction and vibro-impacts; the principle of Saint-Venant; Prandtl models and the gephyroid model; case studies with practical problems in mechanics. 
 
Outline of basic tools for detecting / classifying experimental nonlinear dynamics of mechanical systems; multimodal interactions and regular / non-regular responses of suspended cables in control parameter space; quasiperiodic and homoclinic transitions to chaos; unfolding the complex dynamics triggered by divergence-Hopf bifurcations; identifying and analyzing proper reduced-order models aimed at reproducing experimental scenarios. 
 
Theory of relaxation oscillations, basic concepts, bifurcations, chaos and applications in mechanics; applications of relaxation oscillations to targeted energy transfer problems and to nonlinear vibration isolation designs of discrete coupled oscillators. 
 
Introduction to Analytical Mechanics and application to Multi-Body Dynamics. Generalized coordinates and forces, kinematical constraints, principle of virtual work, Lagrange's equations of motion, integrals of motion, conservation principles. Application to problems in vibration and multibody dynamics; reduced order models in design and study of bifurcations; examples of applications of these techniques to problems encountered in practice.

 

Theory of nonlinear waves and solitons; basic concepts, stability theory and bifurcations of nonlinear waves; methods for studying traveling and standing nonlinear waves; solitary waves as homoclinic points of return maps; applications to problems in mechanics.

 
Course Program
The course will be delivered by recognized professors in the field of nonlinear dynamics: 
 
Bruno Cochelin - Ecole Centrale de Marseille, France
Asymptotic numerical methods for analyzing nonlinear dynamical systems, nonlinear stability analysis, bifurcation theory, aspects of global bifurcations, nonlinear normal modes
 
Vincenzo Gattulli - Universita' dell'Aquila, Italy
Nonlinear dynamical systems: cable and cable-supported beam oscillations, TMD in preventing aereoelastic oscillations
 
Oleg Gendelman - Technion, Israel Institute of Technology, Israel
Theory of relaxation oscillations, bifurcation analysis of oscillators undergoing relaxation oscillations and applications in mechanics 
 
Claude-Henri Lamarque - Ecole Nationale des Travaux Publics de l’Etat, France
Aspects of nonlinear dynamic analysis, identification and advanced signal processing; applications in mechanics, including non-smooth systems with friction and impacts 
 
Angelo Luongo  - Universita' dell'Aquila, Italy
Basics of the method of multiple scales and demonstration of its use for studying bifurcations of free and forced single – and multi – degree of freedom dynamical systems; applications in mechanical systems 
 
Leonid I. Manevitch - Institute of Chemical Physics, Russian Academy of Sciences, Russia
Nonlinear normal modes (NNMs) and nonlinear localization; applications in coupled oscillators 
  
Stéphane Pernot - Ecole Nationale des Travaux Publics de l’Etat, France
Wavelet analysis: identification and advanced signal processing
 
Giuseppe Rega - Universita' di Roma ‘La Sapienza’, Italy
Unfolding experimental nonlinear dynamics and chaotic bifurcation scenarios of suspended cables in the background of theoretical models  
 
Vassilis Rothos - Aristotle University of Thessaloniki, Greece
Nonlinear waves and solitons, basic theory, bifurcation analysis and applications 
 
Stephanos Theodossiades -  University of Loughborough, United Kingdom
Introduction to Analytical Mechanics and application to Multi-Body Dynamics 
 
 
Preliminary Readings
  • G. Kerschen, K. Worden, A.F. Vakakis, J.C. Golinval, Past, present and future of nonlinear system identification in structural dynamics, Mechanical System and Signal Processing 20, 505-592, 2006 
  • K. Worden, G. R. Tomlinson, Nonlinearity in Structural Dynamics: Detection, Identification and Modelling, Institute of Physics Publishing Ltd 2001, ISBN 0 7503 0356 5
  • Guckenheimer J., Holmes P.J., 1983, Nonlinear Oscillators, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag.

  • Rand R.H., 2007, Lecture Notes on Nonlinear Vibrations, downloadable from http://www.tam.cornell.edu/faculty-bio.cfm?NetID=rhr2
  •  Verhulst F., 2003, Nonlinear differential equations and dynamical systems, Springer Verlag. 
  • Wiggins S., 1990, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer Verlag.
 

Hosting Institution

ENTPE Ecole Nationale des Travaux Publics de l'Etat

For further information please contact:

Vincenzo Gattulli - The SICON General Secretary -
Piazzale E. Pontieri 2 - 67040 Monteluco di Roio (Italy)
tel. +39 0862 434511, fax +39 0862 434548
http://www.sicon.ing.univaq.it , e-mail: mailto: This e-mail address is being protected from spam bots, you need JavaScript enabled to view it

Event Time Table

 

Time

Monday

Tuesday

Wednesday

Thursday

Friday

 

March 30

March 31

April 1

April 2

April 3

9:00 ­ 9:45

L.Manevitch

S. Theodossiades

A. Luongo

A. Luongo

V.Rothos

9:45-10:30

L.Manevitch

S. Theodossiades

A. Luongo

A. Luongo

V.Rothos

Coffee Break

11:00-11:45

S. Pernot

S. Pernot

O. Gendelman

V.Rothos

V. Gattulli

11:45-12:30

S. Theodossiades

L.Manevitch

O. Gendelman

V.Rothos

V. Gattulli

Lunch

14:30-15:15

S. Theodossiades

O.Gendelman

G. Rega

V.Rothos

C.H.Lamarque

15:15-16:00

S. Theodossiades

O.Gendelman

G. Rega

O. Gendelman

C.H.Lamarque

Coffee Break

16:30-17:15

C.H.Lamarque

L.Manevitch

B. Cochelin

G.Rega

Discussion/Closure

17:15-18:00

C.H.Lamarque

L.Manevitch

B. Cochelin

G.Rega

 

       

 

 
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