**MATH 950 Spring 2010: **

**Normal Forms of Ordinary Differential Equations**

**Description: **

Given a relatively complicated differential system with a singularity,

there exists, in many instances, a (local) change of coordinates after

which the system takes a most simple form: a normal form. Normal forms

of differential equations are essential tools in the study of

differential equations and in their applications.

We will discuss normal forms in different singular settings, with proofs

of some classical results done in detail (this is important as the

techniques can be adapted to tackle many related problems).

We will discuss normal forms of *vector fields at a singularity* (in the

Poincare domain, in the Siegel domain --and the related diophantine

conditions), normal forms near one regular singularity of the linear

part, of equations with periodic coefficients, the *KAM Theorem* (after

a brief introduction/reminder of Hamiltonian mechanics, and of Liouville

integrability), Fuchsian equations and Boar's theorem (loosely stating

that all equations in mathematical physics can be obtained from a

"master" second order Fuchsian equation).

As time permits (and if there is interest), I will give a brief overview of

recent developments: Gallavotti's correction and Liouville integrability

of Hamiltonian systems, Ecalle's correction and linearization of vector

fields at a resonant singularity, ideas of Ecalle's arborification,

correction and linearization of systems with Fuchsian linear part.

*Prerequisites*: basic ordinary differential equations (theorems of

existence and uniqueness of solutions), basic complex analysis (Taylor

series expansions, convergence criteria).

*Targeted audience*: students studying or using differential equations

will benefit from a solid knowledge of normal forms.

References:

for **normal forms of vector fields**:

V. I. Arnold, *Geometrical Methods in the Theory of Ordinary Differential Equations*, Springer, 1980

Y. Ilyasenko, S. Yakovenko, *Lectures on Analytic Differential Equations* (2007), Graduate Studies in Mathematics, vol. 86

for **hamiltonian systems**:

V. I Arnold, *Mathematical Methods of Classical Mechanics*

G. Gallavotti, *The Elements of Mechanics*, Texts and Monographs in Physics, Springer (July 1983)

*Dynamical Systems *VII (Encyclopaedia of Mathematical Sciences), ed. D.V. Anosov, V.I. Arnold, et.al., Springer 1987

for **Fuchsian systems**:

E. Hille, *Ordinary Differential Equations in the Complex Domain*, Dover Publications (April 9, 1997)

L. Ahlfors, *Complex Analysis*

E. Ince, *Ordinary Differential Equations*, Dover Publications; Repirnt edition (June 1, 1956)

*Dynamical Systems *I (Encyclopaedia of Mathematical Sciences), ed. D.V. Anosov, V.I. Arnold, et.al., Springer 1987

**recent advances:**

J. Ecalle, B. Vallet, *Correction and linearization of resonant vector fields and diffeomorphisms*,

Mathematische Zeitschrift, Vol. 229, No. 2, Oct.1998, 249-318

G. Gallavotti, *A criterion of integrability for perturbed nonresonant harmonic oscillators. ``Wick ordering'' of the perturbations in classical mechanics and invariance of the frequency spectrum*, Comm. Math. Phys. 87, no. 3 (1982), 365-383.

L. H. Eliasson, *Perturbations of stable invariant tori for Hamiltonian systems*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 1, 115--147 (1989).

L. Stolovitch, *Progress in normal form theory*, Nonlinearity 22 (2009)

**A few interesting preprints regarding the KAM Theorem for hamitonian systems**:

About the life and work of Jurgen Moser.

Classical KAM theorem by Jurgen Poschel.

Recent advances in the clasical KAM theorem.

An introduction to the small divisors problem, by Stefano Marmis.

A tutorial on KAM theory, by R. De La Llave.

Developments in the Theory of Hamiltonian Systems, by J. Moser (from SIAM Review).