Deautonomising the Lyness mapping
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912982897786880 |
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| author | Grammaticos, Basil Ramani, Alfred Willox, Ralph |
| author_facet | Grammaticos, Basil Ramani, Alfred Willox, Ralph |
| contents | We examine the Lyness mapping (an integrable $N$th-order discrete system which can be generated from a one-dimensional reduction of the Hirota-Miwa equation) from the point of view of deautonomisation. We show that only the $N=2$ case can be deautonomised when one works with the standard form of the mapping. However it turns out that deautonomisation is possible for arbitrary $N$ when one considers the derivative form of the Lyness mapping. The deautonomisation of the derivative of the $N=2$ case leads to a result we have never met before: the secular dependence in the coefficients of the mapping enters through two different exponential terms instead of just a single one. As a consequence, it turns out that a limit of this multiplicative dependence towards an additive one is possible without modifying the dependent variable. Finally, the analysis of the `late' singularity confinement of the $N=2$ case leads to a novel realisation of the full-deautonomisation principle: the dynamical degree is not given (as is customary) simply by the solution of some linear or multiplicative equation, but is present in the growth of the non-linear (and non-integrable) late-confinement conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24871 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Deautonomising the Lyness mapping Grammaticos, Basil Ramani, Alfred Willox, Ralph Exactly Solvable and Integrable Systems Mathematical Physics We examine the Lyness mapping (an integrable $N$th-order discrete system which can be generated from a one-dimensional reduction of the Hirota-Miwa equation) from the point of view of deautonomisation. We show that only the $N=2$ case can be deautonomised when one works with the standard form of the mapping. However it turns out that deautonomisation is possible for arbitrary $N$ when one considers the derivative form of the Lyness mapping. The deautonomisation of the derivative of the $N=2$ case leads to a result we have never met before: the secular dependence in the coefficients of the mapping enters through two different exponential terms instead of just a single one. As a consequence, it turns out that a limit of this multiplicative dependence towards an additive one is possible without modifying the dependent variable. Finally, the analysis of the `late' singularity confinement of the $N=2$ case leads to a novel realisation of the full-deautonomisation principle: the dynamical degree is not given (as is customary) simply by the solution of some linear or multiplicative equation, but is present in the growth of the non-linear (and non-integrable) late-confinement conditions. |
| title | Deautonomising the Lyness mapping |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2603.24871 |