The idea of continuation methods is the following:
We would like to solve a system of non-linear equations $f(x)=0$ in $\mathbb{R}^n$. Instead of solving the problem directly, we take a look at an easy problem with an obvious solution and trace this solution back to the original problem. This is done by so-called homotopy maps.
For example: $H(x, \lambda) = \lambda f(x)+(1-\lambda)(x-a)$ where $\lambda$ is a scalar and $a \in \mathbb{R}^n$.
My question: What different kind of homotopy maps exists and what are their adventages and disadvantages? My textbook only contains the example above and I also didn't find any information online, expect that it can sometimes be better to use problem-oriented homotopy-maps....