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Consider a system of ordinary differential equations of the form $$ \dot{x}(t) + \frac{1}{t}Ax(t) = Q(x(t)) $$ where $x(t) \in \mathbb{C}^n$, $A \in \mathrm{Mat}_{n\times n}(\mathbb{C})$ is a constant matrix, and $Q: \mathbb{C}^n \to \mathbb{C}^n$ is homogeneous of degree $2$, i.e. $Q(\lambda x) = \lambda^2 Q(x)$ for $\lambda \in \mathbb{C}$.

What is known about existence of solutions near $t = 0$?

If it were not for the quadratic term $Q$, the point $t = 0$ would be a regular singular point of the ODE and then we could use the Frobenius method. But in all the references I know, regular singular points are only discussed for linear systems.

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  • $\begingroup$ mse: math.stackexchange.com/q/3763730/202132 $\endgroup$
    – Simon Parker
    Commented Jul 22, 2020 at 13:21
  • $\begingroup$ By "solutions near $t=0$", do you actually mean solutions analytic/holomorphic at $t=0$ as well ? Do you have any assumptions on the eigenvalues of $A$ ? $\endgroup$
    – Loïc Teyssier
    Commented Jul 22, 2020 at 13:50
  • $\begingroup$ @LoïcTeyssier I'm trying to gather as much information as possible about those systems. So if you can say anything by adding more assumptions (distinct eigenvalues, analyticity, etc), please let me know! $\endgroup$
    – Simon Parker
    Commented Jul 22, 2020 at 13:56
  • $\begingroup$ What I meant is: do you want the solution to extend in any way at $t=0$, or could it be meromorphic or even with an essential singularity? $\endgroup$
    – Loïc Teyssier
    Commented Jul 22, 2020 at 17:02
  • $\begingroup$ @LoïcTeyssier Whatever helps you get an answer. $\endgroup$
    – Simon Parker
    Commented Jul 22, 2020 at 17:21

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There's nothing inherently linear about constructing power series solutions à la Frobenius. The existence and uniqueness theory for a class of singular non-linear ODEs, of which yours is a special case, is treated for instance in Ch.IX of

Wasow, W., Asymptotic expansions for ordinary differential equations, (Dover, 1987) reprint from 1965. ZBL0169.10903

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  • $\begingroup$ Is there a text that focuses less on the existence and uniqueness proof, and more on a recipe for how to construct the solution, please? I'm particularly interested in what to do with the multiple solutions to the indicial equation when one can't rely on any linear combination of solutions to the ODE also being a solution, and how to handle the case where two of the solutions to the indicial equation are degenerate (i.e. same exponent). $\endgroup$
    – user516020
    Commented Nov 2, 2023 at 17:45
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    $\begingroup$ @DanielHatton This sounds like it should be its own question. Please formulate clearly your situation and post it as a question. $\endgroup$
    – Igor Khavkine
    Commented Nov 3, 2023 at 0:16
  • $\begingroup$ Yes, I've been toying with doing that, but the question would probably be more OT on MSE than here. $\endgroup$
    – user516020
    Commented Nov 3, 2023 at 10:28

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