In the course of analysing a particular three dimensional nonlinear dynamical system, I find the need to solve a nonlinear equation of the form:

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

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

The particular equations I'm working with do indeed have parametric solutions, as I was already able to demonstrate a few. However, these solutions assume several simplifying relationships among the parameters $\lambda = (\dots, \lambda_i, \dots)^T$.

Most texts on bifurcation theory, that I looked at, emphasise the methods of characterising different bifurcations at known (i.e., easy to calculate) equilibrium points or invariant manifolds of some dynamical system. The equilibria are found upon solving equations like the one above. The whole gamut of centre manifold theory and normal form theory then follow.

But how are solutions of nonlinear equations such as $\mathcal{M}(x,\lambda)$ found in general?

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

This is clearly a very small set of solutions. Are there any methods to find further solutions?