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.
| Original language | English |
|---|---|
| Pages (from-to) | 723-743 |
| Number of pages | 21 |
| Journal | Journal of Mathematical Sciences |
| Volume | 266 |
| Issue number | 5 |
| DOIs | |
| State | Published - Oct 2022 |
Bibliographical note
Publisher Copyright:© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Funding
The author is grateful to the anonymous referees for numerous and informative remarks, which made it possible to significantly improve the quality of the results obtained in this work.
Keywords
- 15A86
- 90C33
- Polynomial differential systems
- Resolution of singularity
- Topological equivalence
- Vector fields