Linear vs. Nonlinear
Separation of Time Scales
Linear Theory
Simple Bifurcations in Normal Form
Steady Bifurcation
Hopf Bifurcation
Reduction of Dynamics
Center Manifold
Local Representation of Center Manifold
Intervals in Parameter Space
Multiple-Scale Approach
Formal Perturbation Expansion: Solvability
Condition
Steady Patterns
Hopf Bifurcation
Hopf-Hopf Interaction
Takens-Bogdanov Double-Zero
Symmetries
Continuous Symmetries and Slow Dynamics
Phase Diffusion in Taylor Vortex Flow
Fronts Locking of Fronts, Binary-Mixture Convection, Electroconvection in nematics
Perturbed Solitons
Assignments:
Matlab: phaseplane program
pplane5.m,
ppn5out.m
by J.C. Polking
Note: to get the orbits in pplane5 over longer times one can choose ode23 and pick a
calculation window that is 10 times larger than the display window.
XPP-Aut is also often mentioned. I have not used
it yet.
Recommended Books and Papers:
J.D. Crawford, Introduction to Bifurcation Theory, Rev. Mod. Phys. 63 (1991) 991.
P.G. Drazin, Nonlinear Systems,, Cambridge University Press, 1992
call number 515.355 D769n
J. Guckenheimer and P. Holmes, Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields,
Springer, 1983.
call number 519.05 A652 v.42
R. Hoyle Pattern formation : an introduction to methods
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,
Springer 1990. P. Manneville,
Instabilities, chaos and turbulence : an introduction to nonlinear dynamics and complex systems
P. Manneville, Dissipative
Structures and Weak Turbulence, Academic Press, 1990.
M.C. Cross and P.C. Hohenberg,
Pattern Formation Outside of Equilibrium,
Rev. Mod. Phys. 65 (1993) 851.
Most of these books are on reserve the SEL.
call number 501.175 H868p
call number 003.85 W655i
call number 515.39 M282i
call number 531.11 M282d