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The following result in one-dimensional differential equations is sometimes referred to as "Peano phenomenon" (see e.g. here).

If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, the differential equation \begin{align*} \dot{x}=f(x) \end{align*} with initial condition $x(0)=x_0$ can only have more than one solution if $f(x_0)=0$. Is there an analogue of this result in higher dimensions? If there isn't, what is an example of a differential equation with continuous right hand side with multiple solutions starting from a point $x_0$ such that all components of $f(x_0)$ are non-zero?

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  • $\begingroup$ $x'=1, y'=|y|^{1/2}$, $(x,y)(0)=(0,0)$ has non-unique solutions, but the system has no equilibrium points. $\endgroup$
    – Christian Remling
    Commented Sep 24, 2023 at 17:41
  • $\begingroup$ Thank you. I have edited the question to avoid cases like this. I am interested in initial conditions where all components are non-zero. $\endgroup$
    – Todd Chavez
    Commented Sep 24, 2023 at 18:27
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    $\begingroup$ If you change coordinates on Christian Remling's example with a 45-degree rotation, I believe you'll get non-uniqueness at the origin with both components of the derivative nonzero. $\endgroup$
    – Martin M. W.
    Commented Sep 25, 2023 at 12:44

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