From Poincaré–Bendixson theorem follows, that second order differential equations of one variable cannot exhibit chaotic behavior. Does anyone know if the theorem holds for a complex differential equation of the type $$ \ddot{z}=f(\dot{z},z), $$ where $z(t)\in \mathbb{C}$ and $f$ is a holomorphic function? Are all such equations non-chaotic?

  • $\begingroup$ It doesn't look like what I need. The system in the post is 4-dimensional, which means it can be chaotic in principle. Having the restriction of f being holomorphic makes it difficult to find a chaotic system though. $\endgroup$
    – Pavlo. B.
    Commented Mar 9, 2021 at 15:26


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