This an introductory course on dynamical systems, bifurcation theory
and hydrodynamic stability theory. The first part of the course is
devoted to the formulation of the underlying mathematical theory of
bifurcation phenomena arising in parameter-dependent ordinary
differential equations and maps. The second part is devoted to the
study of the fundamental instability mechanisms arising in fluid
flows. The first and second parts of the course will be given by
A. Meseguer and J. Ortin, respectively.
Jordi Ortín .
E-mail: ortinecm.ub.es (UB-Facultat de Física, Office 620)
Course 2010/2011
Syllabus.
Introduction: a review on linear algebra and systems of
linear ordinary differential equations (ODE). Runge-Kutta methods
for numerical integration of ODE. Newton-Raphson methods for
systems of nonlinear algebraic equations. Parameter-dependent
nonlinear systems solving: arclength continuation.
Equilibria of ODE and fixed points of maps: hyperbolicity and
stability properties. Linearization and Hartman-Grossman
theorem. Eigenvalues of linearized ODE and multipliers of
linearized maps. Invariant manifolds: limit cycles.
Non-hyperbolic equilibria and fixed points: center manifold
theory. Center manifold reduction. Normal forms. The homological
equation. Simplification of nonlinear terms.
Parameter-dependent dynamical systems. Bifurcation: definition
and examples. Codimension of a bifurcation. Structural stability
and topological normal form of a bifurcation: transversality and
non-degeneracy conditions.
Codimension-1 bifurcations for ODE: saddle-node (tangent-fold) and
Poincaré-Andronov-Hopf bifurcations.
Codimension-1 bifurcations for maps: saddle-node,
period-doubling (flip) and non-ressonant Naimark-Sacker (torus)
bifurcations.
Stability analysis of limit cycles: Poincaré maps, Floquet
multipliers. Saddle-node, period-doubling and Naimark-Sacker
bifurcations of periodic orbits.
Applications to PDEs: Swift-Hohenberg equation.
Hydrodynamic instabilities (one-fluid systems): Rayleigh's stability theory of jets.
Thermal instabilities: Rayleigh-Bénard convection,
Bénard-Marangoni instability and natural convection. Transition to
the convective state and spatially extended patterns.
Shear intabilities of parallel flows. Plane Couette and
Poiseuille flows. Normal velocity-vorticity formalism:
Orr-Sommerfeld-Squire equations. Accurate numerical computation of
the Orr-Sommerfeld equation. Tollmien-Schlichting waves. Subcritical
transition to turbulence.
References.
Kuznetsov, Y. A., Elements of Applied Bifurcation
Theory. Springer-Verlag, New York, 2004.
Wiggins, S., Introduction to applied nonlinear dynamical
systems and chaos , Springer-Verlag, New York, 1996.
Iooss, G., Adelmeyer, M., Topics in Bifurcation Theory and
Applications , World-Scientific, Singapore, 1999.
Bergé, P., Pomeau, Y., Vidal, C., Order within Chaos:
Towards a Deterministic Approach to Turbulence, Wiley, New
York, 1987.
Drazin, P. G., Introduction to Hydrodynamic Stability ,
Cambridge University Press, Cambridge, 2002.
Guyon, E., Hulin, J. P., Petit, L., Mitescu,
C. D., Physical hydrodynamics,
Oxford University Press, Oxford, 2001.
Joseph, D. D., Stability of fluid motions I &
II Springer-Verlag, Springer Tracts in Natural Philosophy,
27-28, Berlin, 1976.
Manneville, P., Dissipative structures and weak
turbulence, Academic Press, Perspectives in Physics, San Diego,
1990.
fa.upc.edu (UPC-Campus Nord, Dept. Applied
Physics, B5 Bldg., Office 010)