Geometric structure of stationary problem for spatial 1D self-diffusion equation with logistic growth
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911183767863296 |
|---|---|
| author | ICHIDA, Yu |
| author_facet | ICHIDA, Yu |
| contents | This paper considers the solution structure of non-trivial, non-constant stationary states of 1D spatial parabolic equations with nonlinear self-diffusion and logistic growth terms. A two-dimensional ordinary differential equation satisfying the stationary problem is derived and all its dynamics, including to infinity, is revealed by the Poincaré-Lyapunov compactification, one of the compactifications of phase space. The advantage of this method is that it can be used to classify all dynamical systems (especially connecting orbits) of a two-dimensional system including infinity. Therefore, the classification results for the dynamical system including to infinity give the classification results for the non-constant stationary states obtained only from the structure of the original equations. This argument allows us to observe a change in the classification of the non-constant stationary states by an explicit relation between the linear diffusion coefficient and the self-diffusion coefficient, combined with arguments about the symmetries and conserved quantities of the ODEs. This means that changing the self-diffusion coefficient as a bifurcation parameter not only qualitatively changes the dynamical system from a big saddle homoclinic orbit of the ODEs to a heteroclinic orbit that connects the saddle equilibria, but also significantly changes the shape and the properties of the stationary states. It explicitly shows the relationship between linear and self-diffusion, gives a characterization of non-trivial stationary states in terms of dynamical systems, and gives a deep insight into the influence of self-diffusion, one of the nonlinear diffusions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric structure of stationary problem for spatial 1D self-diffusion equation with logistic growth ICHIDA, Yu Dynamical Systems Analysis of PDEs Classical Analysis and ODEs 34C05, 35B40, 35K65 This paper considers the solution structure of non-trivial, non-constant stationary states of 1D spatial parabolic equations with nonlinear self-diffusion and logistic growth terms. A two-dimensional ordinary differential equation satisfying the stationary problem is derived and all its dynamics, including to infinity, is revealed by the Poincaré-Lyapunov compactification, one of the compactifications of phase space. The advantage of this method is that it can be used to classify all dynamical systems (especially connecting orbits) of a two-dimensional system including infinity. Therefore, the classification results for the dynamical system including to infinity give the classification results for the non-constant stationary states obtained only from the structure of the original equations. This argument allows us to observe a change in the classification of the non-constant stationary states by an explicit relation between the linear diffusion coefficient and the self-diffusion coefficient, combined with arguments about the symmetries and conserved quantities of the ODEs. This means that changing the self-diffusion coefficient as a bifurcation parameter not only qualitatively changes the dynamical system from a big saddle homoclinic orbit of the ODEs to a heteroclinic orbit that connects the saddle equilibria, but also significantly changes the shape and the properties of the stationary states. It explicitly shows the relationship between linear and self-diffusion, gives a characterization of non-trivial stationary states in terms of dynamical systems, and gives a deep insight into the influence of self-diffusion, one of the nonlinear diffusions. |
| title | Geometric structure of stationary problem for spatial 1D self-diffusion equation with logistic growth |
| topic | Dynamical Systems Analysis of PDEs Classical Analysis and ODEs 34C05, 35B40, 35K65 |
| url | https://arxiv.org/abs/2509.24752 |