Journal article
Planar chemical reaction systems with algebraic and non-algebraic limit cycles
- Abstract:
- The Hilbert number H(n) is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most n ∈ N. The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, where n is equal to the maximum order of the chemical reactions in the system. Analogues of the Hilbert number H(n) for three different classes of chemical reaction systems are investigated: (i) chemical systems with reactions up to the n-th order; (ii) systems with up to n-molecular chemical reactions; and (iii) weakly reversible chemical reaction networks. In each case (i), (ii) and (iii), the question on the number of limit cycles is considered. Lower bounds on the modified Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve h(x, y) = 0 of degree nh ∈ N and containing one or more ovals in the positive quadrant, a chemical system is constructed which has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the dynamics of the constructed chemical system contain polynomials of degree at most n = 2 nh + 1. Considering nh ≥ 4, the algebraic curve h(x, y) = 0 can contain multiple closed components with the maximum number of ovals given by Harnack’s curve theorem as 1 + (nh − 1)(nh − 2)/2, which is equal to 4 for nh = 4. Algebraic curve h(x, y) = 0 with nh = 4 and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
Actions
Access Document
- Files:
-
-
(Preview, Version of record, pdf, 1.9MB, Terms of use)
-
- Publisher copy:
- 10.1007/s00285-025-02221-0
Authors
- Publisher:
- Springer
- Journal:
- Journal of Mathematical Biology More from this journal
- Volume:
- 90
- Issue:
- 6
- Article number:
- 64
- Publication date:
- 2025-05-22
- Acceptance date:
- 2025-04-11
- DOI:
- EISSN:
-
1432-1416
- ISSN:
-
0303-6812
- Language:
-
English
- Keywords:
- Pubs id:
-
2119162
- Local pid:
-
pubs:2119162
- Deposit date:
-
2025-04-20
- ARK identifier:
Terms of use
- Copyright holder:
- Craciun et al.
- Copyright date:
- 2025
- Rights statement:
- Copyright © 2025, The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
- Licence:
- CC Attribution (CC BY)
If you are the owner of this record, you can report an update to it here: Report update to this record