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In the case of a linear, first-order system with constant coefficients $$ \mathbf{x}'=A \mathbf{x}, $$ with $\mathbf{x} \in \mathbf{R}^n$, it is known that all of the components $x_i$ of $\mathbf{x}$ satisfy the same ODE of order $n$, called the secular equation [at least by Birkhoff and Rota]. It also known that even in the nonlinear case an ODE of high-order can be expressed as a system of first-order ODEs.

My question is: in the case of a nonlinear system of first-order ODEs, is there a method to get a high-order ODE satisfied by the components individually? Special cases, such as polynomial RHS's are also of interest to me.

Thank you!

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  • $\begingroup$ Sure, you just take derivatives and proceed by trying to put terms together, if you can. What happens is that the method proposed by Birkhoff and Rota has no interest whatsoever, certainly not for linear equations (when all is trivial) but it is really nonsense for nonlinear equations in particular because you would need much more regularity (which we know is not needed). $\endgroup$ – John B Nov 3 '18 at 17:27
  • $\begingroup$ @JohnB Thanks John. What do you mean by "if you can"?. Is there some theory on which cases this can be done? $\endgroup$ – user1337 Nov 3 '18 at 17:30
  • $\begingroup$ In the linear case all works well because you can reorganize terms in the form of the so-called secular equation. But in the nonlinear case you almost never can. Just have a look at the first lines of the proof in Birkhoff and Rota and you'll see what I mean. Now change the equation a bit (so that it is nonlinear) and try again. $\endgroup$ – John B Nov 3 '18 at 17:32
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Consider the nonlinear system \begin{align} f_1' &= f_1 f_2 + \alpha_1,\\ f_2' &= -f_1 f_2 +\alpha_2. \end{align} It is possible to massage it into two decoupled second-order ODEs: \begin{align} f_1'' &= f_1 \left(\alpha_1+\alpha _2-f_1' \right)+\frac{f_1' \left(f_1'-\alpha_1\right)}{f_1}, \\ f_2'' &= -f_2 \left(\alpha_1+\alpha_2-f_2'\right)+\frac{f_2' \left(f_2'-\alpha_2\right)}{f_2}. \end{align}

This example suggests that there is no such thing as a nonlinear secular equation, as the equations are different.

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