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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.

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Doesn't the proof for a real variable carry over essentially verbatim, using holomorphicity of the input to ensure path-independence for various integrals and using CR-equations to ensure holomorphicity of the output? – BCnrd May 30 '10 at 17:37
up vote 3 down vote accepted

You'll find relevant information in the book Ordinary differential equations in the complex domain By Einar Hille

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.

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Thank you, this is exactly what I needed – Marco Disce May 30 '10 at 21:02
@Guy Katriel: You're right, the Cauchy-Kowalevski theorem can be applied here but it's probably overkill. – Andrey Rekalo May 30 '10 at 21:27

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.

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Beside the book of late professor Hille I would suggest that you check

Lectures on Analytic Differential Equations by Yulij Iljuashenko and Sergi Yakovenko

Professor Iljuashenko

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. Anosov&field-author=D.V. Anosov&page=1

For starters I think you should check the first volume (if I recall correctly).

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