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!
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.