General multi-scale estimates for Lyapunov data of Perron-Frobenius matrices. The case of diluted autocatalytic chemical reaction networks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917080106795008 |
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| author | Unterberger, Jeremie |
| author_facet | Unterberger, Jeremie |
| contents | Autocatalytic chemical reaction networks are dynamical systems whose linearization around zero, dX/dt = AX, is represented by a Perron-Frobenius matrix A with positive Lyapunov exponent; this exponent gives the growth rate of the species concentration vector X in the diluted regime, i.e. in a vicinity of zero. We introduce here a new, general recursive procedure providing precise quantitative information about Lyapunov data, namely, the Lyapunov eigenvalue, and left and right eigenvectors. Our estimates are based on a multi-scale algorithm inspired from Wilson's renormalization group method in quantum field theory, and Markov chain arguments introduced in (Nghe & Unterberger). They are compatible with the very scarce knowledge of kinetic rates (coefficients of A) generally available in chemistry, and take on the form of simple rational functions of the latter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_11073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General multi-scale estimates for Lyapunov data of Perron-Frobenius matrices. The case of diluted autocatalytic chemical reaction networks Unterberger, Jeremie Probability Mathematical Physics 60J10, 60J27, 60J28, 93-10, 92E20 Autocatalytic chemical reaction networks are dynamical systems whose linearization around zero, dX/dt = AX, is represented by a Perron-Frobenius matrix A with positive Lyapunov exponent; this exponent gives the growth rate of the species concentration vector X in the diluted regime, i.e. in a vicinity of zero. We introduce here a new, general recursive procedure providing precise quantitative information about Lyapunov data, namely, the Lyapunov eigenvalue, and left and right eigenvectors. Our estimates are based on a multi-scale algorithm inspired from Wilson's renormalization group method in quantum field theory, and Markov chain arguments introduced in (Nghe & Unterberger). They are compatible with the very scarce knowledge of kinetic rates (coefficients of A) generally available in chemistry, and take on the form of simple rational functions of the latter. |
| title | General multi-scale estimates for Lyapunov data of Perron-Frobenius matrices. The case of diluted autocatalytic chemical reaction networks |
| topic | Probability Mathematical Physics 60J10, 60J27, 60J28, 93-10, 92E20 |
| url | https://arxiv.org/abs/2511.11073 |