### AM430: Topics in Ordinary Differential Equations

Lectures are Mon: 14-16 room 009   and  Wed. 14-16  room 009

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Topics

Local existence and uniqueness theorems.
Some simple solution methods.
Systems of linear equations
Qualitative analysis
Periodic solutions
Perturbation theory

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Suggested textbooks

Coddington-Levinson Theory of differential equations (McGraw-Hill)
Amann: Ordinary Differential Equations, an introduction to Non Linear Analysis
Tao: Nonlinear Dispersive equations (Chapter one) pdf
Ambrosetti-Prodi: A Primer of Non-linear Analysis
Hirsch-Smale: Differential Equations, Dynamical Systems, and Linear Algebra
For the normal form Bambusi  http://users.mat.unimi.it/users/bambusi/pedagogical.pdf
Presentation of the general case
http://www.scholarpedia.org/article/Normal_forms
For the proof of Hartman Grobman theorem see

Zehnder E. Lectures on dynamical systems.. Hamiltonian vector fields and symplectic capacities.pdf
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Plan of the lectures

Lecture 1-  Mon. 22/2: Plan of the course and introduction
Lecture 2-  Wed. 24/2:  Cauchy-Peano local existence Theorem  (Coddington-Levinson)
Lecture 3-  Mon 29/2:  Contraction mapping theorem, Picard local existence and uniqueness theorem (Tao)
Lecture 4-  Wed 3/3:  Gronwall's lemma, Maximal extensions, global existence theorem (Tao)
Lecture 5- Mon 7/3: Linear systems, fundamental solutions, linear non-homogeneous systems, linear systems with constant coefficients, exponent of a matrix,
Lecture 6-  Wed  9/3:  Jordan form and real canonical form  of a  real matrix (Hirsh-Smale, ask me for the notes), Floquet theorem (complex  form)

Next week we will NOT have class!  remember to read the chapter on normal forms  and do the excercises!!

Lecture 7-  Mon 21/3: Logarithms of real matrices, Floquet theorem (real form)
Lecture 8-  Tue 22/3:  excercises

Lecture 9-  Mon 28/3: Linear systems, sources and wells,  exponential stability and instability results
Lecture 10- Wed 30/3: more on stability and instability results in linear and non-linear setting, Lyapunov functions, Hartmann-Grobmann theorem(statement)
Lecture 11- Mon 4/4: vector fields as derivations, push-forward of a vector field, Lie derivative
Lecture 12- Wed 6/4: more on vector fields and changes of variables the Lie exponentiation formula, symmetries and constants of motion

Next week there are no  classes! please read an introduction on Hamiltonian vector fields and do the excercises!!

Lecture 13-  Wed 20/4: Hamiltonian vector fields, Hamiltonian formalism symplectic changes of variables
Lecture 14-  Wed 27/4: more on Hamiltonian vector fileds and changes of variables
Lecture  15- Fri  29/4:  analytic Hamiltonians, norms and  Cauchy estimates. Polynomial hamiltonians and degree decomposition.
Lecture 16- Mon 2/5:  the basics of Birkhoff normal form. One dimensional  case, geometric interpretation.
Lecture 17-  Wed 4/5: More on Birkhoff normal form sequences of changes of variables and estimates
Lecture 19-  Mon 9/5: Statement of the Birkhoff theorem in C^k class,  the general alogrithm
Lecture 20-  Wed 11/5: The elliptic non-resonant case.  Reducibility algorithms, a firs example diagonalization of regular matrices, a perturbative approach.

Lecture 21-  Mon 16/5:
Lecture 22-  Wed 18/5:

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Exercises:

1.  Prove the Cauchy-Kovaleskii Theorem (Exercise 1.1 Tao )
2.  Exercise 1.7, 1.10 and 1.11  in Tao. (Compartison principle and existence times)
3. Discuss global existence for  one dimensional autonomous systems $\dot x = f(x)$ where $f$ is a C^1 function $R\to R$.
4. Exercises on linear systems pdf
5. Prove Floquet theorem in real form: Suppose that we have a real ODE \dot x = A(t) x, with $A(t)$ a real $T$-periodic matrix. Prove that there exists a real $2T$-periodic change of variables which reduces  A(t) to constant coefficients.
6. Let $r(x,\tau)$ be the push-forward of the vector field $f(y)$ w.r.t. the flow $y= \Phi(\tau,x)$ of  $\frac {d y}{d\tau} = g(y)$, $y(0)= x$.

Write an explicit formula for $r$ in terms of $\Phi$.

Show that $\partial_\tau r = [r,g]$. Hint: prove it first for $\tau=0$.

Suppose now that $g(y)$ is a Hamiltonian vector field with Hamiltonian $S(p,q)$  (so $y=(p,q)$ and $x=(P,Q)$). Given any Hamiltonian $H(y)$ let $K(\tau,x):= H(\Phi(\tau,x)) (recall that$y=\Phi(\tau,x)$is the flow. Show that$\partial_\tau K(\tau,x)= \{K(\tau,x),S(x)\}$. 7. Consider the space of analytic functions$H(p,q)= \sum_{a,b\in \mathbb N_0} H_{a,b} p^a q^b$where$H_{a,b}$is a sequence of real numbers such that $$|H|_r:= \sum_{a,b\in \mathbb N_0} |H_{a,b}|r^{a+b}<\infty.$$ Show that for all$r'<r$one has the Cauchy estimates $$|\{H,K\}|_{r'} \le C(r,r')|H|_r |K|_r$$ give an estimate for$C(r,r')$as for$r'$close to$r$. 8. Consider the space of regular analytic Hamiltonians$H(p,q)= \sum_{a,b\in \mathbb N_0^d} H_{a,b} p^a q^b$where$H_{a,b}$is a sequence of real numbers such that $$|H|_r:= \sum_{a,b\in \mathbb N_0}(1+|a|+|b|) |H_{a,b}|r^{a+b-2}<\infty.$$ or alternatively $$|H|_r= \frac{1}{r} \sup_{|p|,|q|\le r, p,q\in \C^d} |X_H(p,q)|$$ prove Cauchy estimates $$|\{H,K\}|_{r'} \le C(1-r'/r)^{-1}|H|_r |K|_r$$ discuss the existence times for$e^{t\{F,\cdot\}}$as a map from regular ananytic functions in B(r') to regular analytic functions in B(r). 9. Discuss the dynamical consequences of the non-resonant elliptic Birkhoff normal form theorem (see notes by Bambusi) 10. compute two steps of Birkhoff normal form for the Hamiltonian $$1/2(p_1^2+q_1^2) + a/2 (p_2^2+q_2^2)+ p_1p_2^2+ q_1^2 p_1$$ for$a= 1$and$a= \sqrt{2}\$.