Analytic ODE with complex time - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:01:23Z http://mathoverflow.net/feeds/question/26474 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26474/analytic-ode-with-complex-time Analytic ODE with complex time Marco 2010-05-30T17:30:19Z 2010-07-11T08:42:51Z <p>Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| &lt; L$ on ball $B_r$ with radius r. I would like to understand:</p> <p>1) if there exists an analytic flow $\phi_t(x)$ with complex time $t$ such that $\partial_t \phi_t(x)=v(\phi_t(x))$</p> <p>2) if such flow is analytic in both $t$ and $x$</p> <p>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).</p> <p>Are there references about this kind of problems?</p> <p>Thank you for your attention.</p> http://mathoverflow.net/questions/26474/analytic-ode-with-complex-time/26502#26502 Answer by Guy Katriel for Analytic ODE with complex time Guy Katriel 2010-05-30T20:40:43Z 2010-05-30T20:40:43Z <p>You'll find relevant information in the book Ordinary differential equations in the complex domain By Einar Hille</p> <p><a href="http://books.google.com/books?id=I1OR4t8UZCgC&amp;printsec=frontcover&amp;dq=Ordinary+Differential+Equations+in+the+Complex+Domain&amp;ei=t8wCTPaXFZLKygT_9e24DA&amp;cd=1#v=onepage&amp;q&amp;f=false" rel="nofollow">http://books.google.com/books?id=I1OR4t8UZCgC&amp;printsec=frontcover&amp;dq=Ordinary+Differential+Equations+in+the+Complex+Domain&amp;ei=t8wCTPaXFZLKygT_9e24DA&amp;cd=1#v=onepage&amp;q&amp;f=false</a></p> <p>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.</p> http://mathoverflow.net/questions/26474/analytic-ode-with-complex-time/26513#26513 Answer by Daniel Asimov for Analytic ODE with complex time Daniel Asimov 2010-05-30T22:38:26Z 2010-05-30T22:38:26Z <p>One thing to be careful about is that even for an analytic ODE given on ℂ via</p> <p>dz/dt = f(z)</p> <p>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) . . .</p> <p>. . . <strong>but</strong> 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 </p> <p>O<sub>0</sub> x (-K,K) and </p> <p>O<sub>1</sub> x (-K,K) </p> <p>in ℂ x ℝ such that for some t<sub>0</sub> in (-K,K) we have, e.g.,</p> <p>Φ(z<sub>1</sub>,t<sub>0</sub>) = z<sub>1</sub> for all z<sub>1</sub> ∈ O<sub>1</sub>, although</p> <p>Φ(z<sub>0</sub>,t<sub>0</sub>) ≠ z<sub>0</sub> for all z<sub>0</sub> ∈ O<sub>0</sub>.</p> <p>This seems to violate permanence, but for a subtle reason does not.</p> <p>For a concrete example: let P(z) := i(z<sup>3</sup> - z), and let O<sub>j</sub> be a small open neighborhood of j in ℂ. Then the flow given by Φ(z,t) := z(t) satisfying</p> <p>dz/dt = P(z)</p> <p>is defined for all real time t on both O<sub>0</sub> and O<sub>1</sub>.</p> <p>But setting t<sub>0</sub> = &pi;, we have</p> <p>Φ(z,&pi;) - z = 0 for all z in O<sub>1</sub>, although</p> <p>Φ(z,&pi;) - z ≠ 0 for all z in O<sub>0</sub>.</p> http://mathoverflow.net/questions/26474/analytic-ode-with-complex-time/31377#31377 Answer by Predrag Punosevac for Analytic ODE with complex time Predrag Punosevac 2010-07-11T08:42:51Z 2010-07-11T08:42:51Z <p>Beside the book of late professor Hille I would suggest that you check </p> <p>Lectures on Analytic Differential Equations by Yulij Iljuashenko and Sergi Yakovenko</p> <p><a href="http://www.amazon.com/Lectures-Analytic-Differential-Equations-Mathematics/dp/0821836676" rel="nofollow">http://www.amazon.com/Lectures-Analytic-Differential-Equations-Mathematics/dp/0821836676</a></p> <p>Professor Iljuashenko </p> <p><a href="http://www.math.cornell.edu/People/Faculty/ilyashenko.html" rel="nofollow">http://www.math.cornell.edu/People/Faculty/ilyashenko.html</a></p> <p>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.</p> <p>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.</p> <p><a href="http://www.amazon.com/s?ie=UTF8&amp;rh=i%3Astripbooks%2Cp_27%3AD.V" rel="nofollow">http://www.amazon.com/s?ie=UTF8&amp;rh=i%3Astripbooks%2Cp_27%3AD.V</a>. Anosov&amp;field-author=D.V. Anosov&amp;page=1</p> <p>For starters I think you should check the first volume (if I recall correctly).</p>