Analytic ODE with complex time

Question

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.

Answer 1

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.

Answer 2

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 O_j 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.

Answer 3

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).