MATH-6400, Ordinary Differential Equations and Dynamical Systems

MATH-640

ORDINARY DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS


Time: Monday, Thursday, 12:00 to 1:50 PM
Room: Carnegie 106
Instructor: Gregor Kovacic
Office: 419 Amos Eaton
Phone: 276-6908
E-mail: kovacg at rpi dot edu
Office Hours: Click here.


This document contains the list of topics to be presented in this course, list of homework problems, a list of textbooks that contain some of the course topics, and some comments and recommendations about these textbooks.


Topics

Topics in this course will be selected from among the following:

Existence and uniqueness: contraction mappings, Picard approximations, Lipschitz condition, local existence and uniqueness, continuous dependence and differentiability with respect to parameters, rectification, extension of solutions, equation of variations.

2nd-order differential equations with analytic coefficients: Ordinary points and regular singular points, convergent series expansions, classification according to the number of regular singular points, hypergeometric equation, irregular singular points, asymptotic expansions of solutions, equations with a large parameter, WKB method.

Equilibrium points and their stability: Lyapunov stability, linearization, stable manifold theorem, contraction mapping and Perron's method.

Periodic solutions and Poincare maps: Surface of section, Poincare return map, Floquet theory, stability of periodic orbits.

Local bifurcations of equilibria: saddle-node, transcritical, pitchfork, period doubling, Hopf, center manifolds, normal forms.

Perturbation methods: Gronwall's inequality, averaging, subharmonic and homoclinic Melnikov methods, obstructions to averaging, resonances.

Global bifurcations and chaos: Smale horseshoes, symbolic dynamics, chaos, transverse homoclinic orbits, lambda lemma, Smale-Birkhoff homoclinic theorem.

Sturn-Liouville theory: Sturm's oscillation and comparison theorems, self-adjoint and non-self-adjoint equations, Green's functions, eigenfunction expansions, non-self-adjoint problems and expansions in bi-orthogonal eigenfunctions, singular eigenvalue problems.

Integrable partial differential equations: Scattering and inverse scattering, Korteweg-deVries equation, Lax pair, inverse-scattering transform, Gelfand-Levitan-Marchenko equation, conservation laws.

Equations of neuroscience: Hodgkin-Huxley equations and derivation, integrate-and-fire model, computational methods, Fokker-Planck and kinetic-theoretic coarse-grained description, mean-field approximation, neuronal network oscillations.

The class notes are deposited here.


Homework Problems

Click here for a list of homework problems and weekly assignments.


Textbooks

The following textbooks contain some of the material similar to that presented in the course:

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge University Press
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM
D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University Press
V. I. Arnold, Ordinary Differential Equations, MIT press or Springer Verlag.
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer Verlag.
V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations, Springer Verlag.
G. D. Birkhoff, Dynamical Systems, AMS Publications.
G. Birkhoff and G. Rota, Ordinary Differential Equations, Wiley.
F. Brauer and J. A. Nohel, Ordinary Differential Equations, Benjamin.
F. Brauer and J. A. Nohel, Qualitative Theory of Ordinary Differential Equations, Benjamin.
C. Chicone, Ordinary Differential Equations with Applications, Springer.
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger.
R. Courant and D. Hilbert, Methods of Mathematical Physics, Part I, Wiley-Interscience.
P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT Press.
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin-Cummings.
G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, Springer.
W. Gerstner and W. M. Kistler, Spiking Neuron Models: Single Neurons, Populations, Plasticity, Cambridge University Press.
E. Hille, Differential Equations in the Complex Domain, Dover.
J. K. Hale, Ordinary Differential Equations, Dover.
M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press.
J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag.
E. L. Ince, Ordinary Differential Equations, Dover.
E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press.
C. Koch, Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press.
G. L. Lamb, Elements of Soliton Theory, Wiley
J. Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press.
A. C. Newell, Solitons in Mathematics and Physics, SIAM
S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons, The Inverse Scattering Method, Plenum Publishing Corporation
F. W. J. Olver, Asymptotics and Special Functions, CRC Press
I. Percival and D. Richards, Introduction to Dynamics, Cambridge University Press.
D. A. Sanchez, Ordinary Differential Equations and Stability Theory: an Introduction, Dover.
J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer Verlag.
H. C. Tuckwell, Introduction to Theoretical Neurobiology: Volume 1, Linear Cable Theory and Dendritic Structure; Volume 2, Nonlinear and Stochastic Theories, Cambridge University Press.
W. Walter, Ordinary Differential Equations, Springer Verlag
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer Verlag.


Comments and recommendations about textbooks

I only ordered Arnold's ODEs and Coddington and Levinson to be on sale in the RPI bookstore, and even these two books only as optional. Arnold's ODEs is a great little book written by a great mathematician, so I strongly recommend buying it to everybody. If you want the classic texbook on the subject and a good standard reference book, buy the book by Coddington and Levinson. It contains most of the relevant knowledge about ODEs in the late 1950s, and most of its contents are still relevant today. More modern, but also harder, is the book by Hale, which was state-of-the art ODEs with a strong dynamical systems leaning in the 1970s. Arnold's Classical Mechanics is another classic, containing a great exposition of mechanics from a modern viewpoint. Its contents range from Newton's law to differential forms and symplectic geometry. It has a number of great appendices, each presenting a "crash course" on an important topic in dynamical systems. If you want a simple, painless introduction to Lagrangian and Hamiltonian Dynamics, take a look at the book by Percival and Richards. Arnold's Geometric Methods book delves deeper into the dynamical systems theory. Moser's book covers some classic dynamical systems topics that were the subject of intense research from 1960's to 90s. The book by Olver contains the material on the asymptotic behavior of 2nd order ODEs near irregular singular points and with large parameters, which is more accessible than that in Coddington and Levinson. The books by Hirsch and Smale, Birkhoff and Rota, Brauer and Nohel, Hille, Sanchez, and Walter are good standard textbooks containig some of the material covered in this course. I kept some of the dynamical systems books from the 80s and early 90s, which are a bit dated by now, but they contain some topics that cannot be found elsewhere. Of the books on Soliton Theory, the most accessible is the one by Lamb. The standard text now would be the one by Ablowitz and Clarkson. I added a number of standard texbooks on theoretical neuroscience. My favorite is the one by Koch. I also added two books on mathematical neuroscience, which emphasize connections with dynamical systems.


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