In the course of analyzing a particular nonlinear three-dimensional dynamic system, I find the need to solve a non-linear equation of the form:

$$ mathcal {M} (x, lambda): = x – f (x, lambda_1, dots, lambda_p) = 0 $$

With $ x in mathbb {R} ^ 3 _ + $ Y $ f colon Omega rightarrow mathbb {R} ^ 3 $Y $ Omega times Lambda subset mathbb {R} ^ 3_ + times mathbb {R} ^ p _ + $, open.

The particular equations with which I am working have parametric solutions, as I have already demonstrated some. However, these solutions assume several simplifying relationships between the parameters. $ lambda = ( dots, lambda_i, dots) ^ T $.

Most of the texts on bifurcation theory that I observed emphasize the methods of characterizing different bifurcations at known points of equilibrium (that is, easy to calculate) or multiple invariants of some dynamic system. Balances are found by solving equations like the previous one. The whole range of the theory of the multiple center and the theory of the normal form follow below.

But what are the solutions of nonlinear equations like? $ mathcal {M} (x, lambda) $ found in general?

Here it may be relevant to point out how I found solutions for the particular equations in which I am working. If we think of $ x $ as $ (x_1, x_2, x_3) ^ T $, after the elimination of $ x_2, x_3 $ From the system, I was left with a high-grade polynomial in $ x_1 $ and then reduced the grade by making some assumptions between the parameters.

This is clearly a very small set of solutions. Is there any method to find more solutions?