Analytic ODE with complex time Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi_t(x)$ with complex time $t$ such that $\partial_t \phi_t(x)=v(\phi_t(x))$
2) if such flow is analytic in both $t$ and $x$
3) if the domain of the variable $t$ where $\phi_t(x)$ is analytic is bounded by $r (\sup_{B_r} |v|)^{-1}$ (or something like that).
Are there references about this kind of problems?
Thank you for your attention.
 A: You'll find relevant information in the book
Ordinary differential equations in the complex domain
By Einar Hille
http://books.google.com/books?id=I1OR4t8UZCgC&printsec=frontcover&dq=Ordinary+Differential+Equations+in+the+Complex+Domain&ei=t8wCTPaXFZLKygT_9e24DA&cd=1#v=onepage&q&f=false
Fixed point (iteration) results are used to prove local existence, and also to give explicit 
lower bounds on the domain of existence, like you want.
A: One thing to be careful about is that even for an analytic ODE given on ℂ via
dz/dt = f(z)
where f is an entire function, the solutions Φ(z,t) always exist for all (z,t) in some open neighborhood of ℂ x {0} in ℂ x ℂ or just ℂ x ℝ (if we just consider real time) . . .
. . . but even for f(z) = a mere polynomial P(z), a paradoxical phenomenon can occur, even just considering real time: The flow Φ(z,t) can be defined, for some K > 0, on two disjoint open sets 
O0 x (-K,K) and 
O1 x (-K,K) 
in ℂ x ℝ such that  for some t0 in (-K,K) we have, e.g.,
Φ(z1,t0) = z1 for all z1 ∈ O1, although
Φ(z0,t0) ≠ z0 for all z0 ∈ O0.
This seems to violate permanence, but for a subtle reason does not.
For a concrete example: let P(z) := i(z3 - z), and let Oj be a small open neighborhood of j in ℂ.  Then the flow given by Φ(z,t) := z(t) satisfying
dz/dt = P(z)
is defined for all real time t on both O0 and O1.
But setting t0 = π, we have
Φ(z,π) - z = 0 for all z in O1, although
Φ(z,π) - z ≠ 0 for all z in O0.
A: Beside the book of late professor Hille I would suggest that you check 
Lectures on Analytic Differential Equations by Yulij Iljuashenko and Sergi Yakovenko
http://www.amazon.com/Lectures-Analytic-Differential-Equations-Mathematics/dp/0821836676
Professor Iljuashenko 
http://www.math.cornell.edu/People/Faculty/ilyashenko.html
who is now at Cornell has delivered many times courses on differential 
equations in complex domain back in Moscow and is one of the greatest experts in the field.
As with any question on dynamical systems I would also suggest that you have handy nine volumes of Encyclopaedia of Mathematical Sciences dedicated to Dynamical Systems. The series is edited by Arnold, Anosov, Sinai, Novikov, and few other outstanding Soviet mathematicians.
http://www.amazon.com/s?ie=UTF8&rh=i%3Astripbooks%2Cp_27%3AD.V. Anosov&field-author=D.V. Anosov&page=1
For starters I think you should check the first volume (if I recall correctly).
