Elementary catastrophes underlying bifurcations of vector fields and PDEs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911909604753408 |
|---|---|
| author | Jeffrey, Mike R |
| author_facet | Jeffrey, Mike R |
| contents | A practical method was proposed recently for finding local bifurcation points in an n-dimensional vector field F by seeking their 'underlying catastrophes'. Here we apply the idea to the homogeneous steady states of a partial differential equation as an example of the role that catastrophes can play in reaction diffusion. What are these 'underlying' catastrophes? We then show they essentially define a restricted class of 'solvable' rather than 'all classifiable' singularities, by identifying degenerate zeros of a vector field F without taking into account its vectorial character. As a result they are defined by a minimal set of r analytic conditions that provide a practical means to solve for them, and a huge reduction from the calculations needed to classify a singularity, which we will also enumerate here. In this way, underlying catastrophes seem to allow us apply Thom's elementary catastrophes in much broader contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14818 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Elementary catastrophes underlying bifurcations of vector fields and PDEs Jeffrey, Mike R Mathematical Physics Dynamical Systems 34Cxx, 37Nxx, 35Axx, 53Zxx A practical method was proposed recently for finding local bifurcation points in an n-dimensional vector field F by seeking their 'underlying catastrophes'. Here we apply the idea to the homogeneous steady states of a partial differential equation as an example of the role that catastrophes can play in reaction diffusion. What are these 'underlying' catastrophes? We then show they essentially define a restricted class of 'solvable' rather than 'all classifiable' singularities, by identifying degenerate zeros of a vector field F without taking into account its vectorial character. As a result they are defined by a minimal set of r analytic conditions that provide a practical means to solve for them, and a huge reduction from the calculations needed to classify a singularity, which we will also enumerate here. In this way, underlying catastrophes seem to allow us apply Thom's elementary catastrophes in much broader contexts. |
| title | Elementary catastrophes underlying bifurcations of vector fields and PDEs |
| topic | Mathematical Physics Dynamical Systems 34Cxx, 37Nxx, 35Axx, 53Zxx |
| url | https://arxiv.org/abs/2310.14818 |