Abstract
An explicit method, based on subsequent small perturbations, allowing one to study the algebraic and geometric nature of multiple isolated singularities of a polynomial vector field, is discussed. The main ingredients of the method are (i) establishing a canonical form of a singularity, (ii) explicit decomposition of a compound singularity into simpler ones, and (iii) deriving asymptotic laws of decomposition/collision of singularities. In particular, the saddle-node, pitchfork, and quadruple bifurcations of zeros of a polynomial vector field are considered from the various novel and perhaps unexpected angles. Several examples of subsequent phase portraits illustrating possible interactions between equilibrium of ODEs are also discussed.
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Appendix
Appendix
We start first with phase portraits of a saddle-node bifurcation, that is, in fact, a collision and disappearance of two equilibria in dynamical systems generated by autonomous ODEs,
Examples of a system with a semi-hyperbolic singularity
After splitting the point, one obtains two simple points: a node and a saddle. The absolute values of the Jacobian determinants are equal (in general, asymptotically identical). The signs of the determinants, however, are always opposite.
An example of a system with a nilpotent singularity
Splitting a singularity of multiplicity 2 along a straight line into two simple, singular points, it is easy to see that the type of critical point is a cusp tip, which is a combination of the simple singular points: a center and a saddle (Fig. 7).
Saddle — center splitting
For different systems, whose \(\overrightarrow{{p}}\left( x_1\right)\) vectors are identical, one can see a change in the type of the critical point splitting. In the first, the point splits into a saddle and a focus. In the second system, the point splits — into a saddle and a center (see Fig. 8) (although both systems are obtained from imaginary eigenvalues). This phenomenon is known as the center-focus problem, as written in [10] (Fig. 9).
Saddle — focus splitting
Splitting of a multiple singularity of multiplicity 3 to three points along a straight line trajectory
Decomposition of singularity of multiplicity 4
We presented three different scenarios of possible splitting of an isolated singularity of multiplicity 4 [5] in Figs. 10 and 11.
The singular points are at the vertices of a triangle, with one point in the middle
Splitting of the multiple singularity with a multiplicity of 4 to four simple points located at the convex rectangular vertices’s
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Rabinowitz, I. STEP-BY-STEP RESOLUTION OF SINGULARITIES AND STUDYING INTERACTIONS BETWEEN THEM. J Math Sci 266, 723–743 (2022). https://doi.org/10.1007/s10958-022-06056-8
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DOI: https://doi.org/10.1007/s10958-022-06056-8