Marie Curie Europe

TC1: Stability and Bifurcations of Nonlinear Dynamical Systems

TC1: Stability and Bifurcations of Nonlinear Dynamical Systems

A training course coordinated by Angelo Luongo
DISAT - University of L’Aquila, 2-6 July 2007

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This course will illustrate the basic principles of Dynamical System Theory, as well as the operating tools to solve practical problems in several engineering branches. Considering a system as a vector field or a map, the notions of equilibrium points, periodic and quasiperiodic motions will be introduced. Then, the fundamental concepts of bifurcation and stability will be given for a multiparameter system, and the different types of loss of stability will be discussed (divergence, Hopf, Period-Doubling, secondary Hopf). Related notions as control parameters, transversality condition, codimension, Poincarè maps, linear stability diagrams and bifurcation diagrams will be given. The multi-parameter bifurcation theory of eigenvalues, which is a key point for stability and instability studies, will be addressed. The singularities of stability boundaries will be analyzed and a consistent explanation given for several interesting mechanical effects, like gyroscopic stabilization, flutter and divergence instabilities, transference of instability between eigenvalue branches, destabilization and stabilization by small damping, disappearance of flutter instability and parametric resonance in periodically excited systems.

Two alternative classes of operating methods, both applied in the literature, will be discussed. In the first approach, the Center Manifold Theory, the Lyapunov-Schmidt method and the Normal Form Theory, will be illustrated as methods able to reduce the original system to a smaller dimensional system, incorporating the essential system dynamics. A set of examples taken from mechanics and engineering will show the usefulness of aforementioned methods. These examples include buckling problems of rods, plates and shells, the loss of stability of the motion of vehicles, of simple robots, and elastic tubes conveying fluid. With these examples, issues like symmetry-breaking, pattern formation, imperfection sensitivity and correct modeling of systems will be explored.

As a second, more recent and engineering-oriented approach, the Multiple Scale Perturbation Method and the Lindstedt-Poincarè Method, well known in Nonlinear Dynamics community, will be illustrated in detail. The approach appears to be a natural extension to dynamical systems of the ‘static perturbation method’, commonly applied in the literature since the 60’s in the buckling analysis framework. The algorithms call for some knowledge of perturbation methods, linear algebra, and eigenvalue sensitivity. Many applications to bifurcations from a known path of equilibrium points (as divergence, resonant and non-resonant Hopf bifurcations and their combinations) or of periodic motions (divergence, period-doubling and secondary Hopf bifurcations) will be studied, both for discrete and continuous systems. Many mechanical examples will be discussed, and numerical results shown. Aeroelastic problems as beams or cables under wind flows will be studied; moreover the effects of added small masses (Tuned Mass Dampers) on the system stability will be analyzed, as significant example of passive control.
Finally, the effects of nonlinearities on parametrically-excited systems and the stability of systems governed by differential delayed equations will be studied.

Invited Lecturers

H. Troger – Wien University of Technology, Austria
8 lectures on: Definition of dynamical systems (discrete and continuous) and basic concepts; Dimension reduction (Lyapunov-Schmidt, Center Manifold, Inertial Manifold; Nonlinear Galerkin), Normal form reduction; Bifurcation in symmetric system; Classification of bifurcations with low codimensions and their presentation in diagrams.

A. Seyranian – Moscow State Lomonosov University, Russia
8 lectures on: Bifurcation analysis of eigenvalues, left eigenvectors, Jordan chains; Perturbation of simple and multiple eigenvalues; Stability boundaries of multiparameter-dependent systems; Stability of Hamiltonian, damped Hamiltonian and gyroscopic systems; Stability of periodic systems, Floquet theory, Mathieu equation.

A. Luongo – University of L’Aquila, Italy
8 lectures on: Introduction to Perturbation Methods; Static Perturbation Method for buckling problems; Multiple Scale Method for static and/or dynamic bifurcations; Multiple bifurcations from a known path of discrete and continuous systems; Asymptotic Methods for the analysis of non-autonomous periodic systems.

R. Rand – Cornell University, Ithaca NY, USA
8 lectures on: Nonlinear parametric excitation; Differential delay equations.

A. Steindl – Wien University of Technology, Austria

4 lectures on: Numerical calculation of branching behaviour (e.g. AUTO, Matcont); Applications to all typical low codimension cases for algebraic equations and ordinary differential equations; Applications in finite and infinite dimensional cases.

A. Di Egidio – University of L'Aquila, Italy
2 lectures on: Applications of the Multiple Scale Methods to sample mechanical systems.

Preliminary Readings

Troger, H. and Steindl, A., Nonlinear Stability and Bifurcation Theory, Springer Verlag, Wien, New York, 1991.

Seyranian, A.P. and Mailybaev, A.A., Multiparameter Stability Theory with Mechanical Applications, World Scientific, New Jersey, 2003.

Rand, R., Lecture Notes on Nonlinear Vibrations.

Luongo, A., Di Egidio, A., Paolone, A., Multiple Scale Bifurcation Analysis for Finite-Dimensional Autonomous Systems in ‘Recent Research Developments in Sound & Vibration’, Transworld Research Network, Kerala, India, ISBN:81-7895-031-6 161-201, 2002.

Hosting Institution

DISAT is the Department of Structural, Hydraulic and Geotechnical Engineering at the University of L’Aquila. It is located in Monteluco hill, which dominates L’Aquila, a middle-size town in the centre of Italy, in the Abruzzo region. A beatiful view of the Gran Sasso, the highest peak of the Appennini mountains, non-polluted air and a pine forest characterize its location.

For further information please contact:

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


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