If we have a dynamical system that is not in general position (see Shilnikov, 2nd book, p. 546) that means the matrix:
[ \frac{\partial G}{\partial \epsilon_1} & \ldots & \frac{\partial G}{\partial \epsilon_p} \\
\vdots & & \vdots \\
\frac{\partial^{k-1} G}{\partial \epsilon_1 \partial x^{k-2}} & \ldots &
\frac{\partial^{k-1} G}{\partial \epsilon_p\partial x^{k-2}} ]
Where $G$ is the system restricted to the center manifold, $\epsilon$ are $p$ parameters, $x$ is the variable and the $k$-st Lyapunov value doesn't vanish (but all
@Jochen - your question seems to be cut off part way through. The basic answer is, 'if you don't assume general position then anything can happen'! But of course that's not very helpful. In my experience, there is usually a reason that general position fails. One such reason is that the system has symmetry, and another is that one knows for some a priori reason, that $x=0$ is a solution for all $\epsilon$. If there is a (mathematical) reason for the failure of general position, then one can usually incorporate it into the bifurcation theory and determine what the most likely bifurcations are (though that's not always straightforward).
For the symmetry case, much can be found in the book of Golubitsky, Stewart and Schaeffer. For example, with a reflection symmetry one sees the pitchfork with just 1 parameter.
This is an interesting question with more than one possible answer.
A good place to start in looking for an answer to this question is
T. Lewiner, C.O.L. Ferreira, M. Craizer, R. Teixeira, Curvature motion for union of balls, 2005:
http://www.matmidia.mat.puc-rio.br/tomlew/pdfs/medialmove_sibgrapi.pdf
See Section 3 (Curvature motion on the medial axis), starting on page 3. This is interesting (for me) because these authors consider Vorono\"i diagrams, Delaunay triangulation and medial axes when the points are not in the general position.
Bifurcation in systems not in the general position is considered in
See Theorem H.1, page 341.
Vorono\"i tessellation of regions of infinite extent with borders that are parallel (not in the general position) are considered briefly in
A.D.V. Govea, Incremental learning for motion prediction of pedestrians and vehicles, Ph.D. thesis, Institut National Polytechnique de Grenoble, 2007:
http://emotion.inrialpes.fr/fraichard/publications/internships/07-phd-vasquez.pdf
See, for example, Section 6.5.3, starting on page 86.
@Jochen - your question seems to be cut off part way through. The basic answer is, 'if you don't assume general position then anything can happen'! But of course that's not very helpful. In my experience, there is usually a reason that general position fails. One such reason is that the system has symmetry, and another is that one knows for some a priori reason, that $x=0$ is a solution for all $\epsilon$. If there is a (mathematical) reason for the failure of general position, then one can usually incorporate it into the bifurcation theory and determine what the most likely bifurcations are (though that's not always straightforward).
For the symmetry case, much can be found in the book of Golubitsky, Stewart and Schaeffer. For example, with a reflection symmetry one sees the pitchfork with just 1 parameter.
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