Noise-induced stabilization of saddle-node ghosts
It is known that saddle-node (s-n) bifurcations leave a saddle remnant (or ghost) in the region of the phase space where the annihilation of the fixed points occurred. The corresponding time delay, t _d , found right after the bifurcation is known to follow the scaling law ${t}_{\mathrm{d}}\sim {\le...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2020-01-01
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| Series: | New Journal of Physics |
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| Online Access: | https://doi.org/10.1088/1367-2630/abb549 |
| _version_ | 1797750356417970176 |
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| author | Josep Sardanyés Carles Raich Tomás Alarcón |
| author_facet | Josep Sardanyés Carles Raich Tomás Alarcón |
| author_sort | Josep Sardanyés |
| collection | DOAJ |
| description | It is known that saddle-node (s-n) bifurcations leave a saddle remnant (or ghost) in the region of the phase space where the annihilation of the fixed points occurred. The corresponding time delay, t _d , found right after the bifurcation is known to follow the scaling law ${t}_{\mathrm{d}}\sim {\left({\epsilon}-{{\epsilon}}_{\mathrm{c}}\right)}^{-1/2}$ , where ϵ and ϵ _c are the control parameter and its critical value, respectively. While the properties of such delays are well understood for deterministic systems, much less is known about how intrinsic noise influences this phenomenon. As a first step towards analysing this issue, in this article we explore a model with autocatalysis and a two-species hypercycle to analyse the impact of noise on delayed transitions in one- and two-dimensional dynamical systems suffering a s-n bifurcation. The first model is investigated with Gillespie simulations and the diffusion approximation, focusing on the behaviour and properties close to the bifurcation. A Fokker–Planck equation is derived, together with the stochastic potential. We show that the slowing down of the dynamics remains robust to noise. In fact, we prove both analytically and numerically that increasing noise lengthens the delays after bifurcation threshold. Furthermore, the inverse square-root scaling law is not robust to fluctuations. By contrast, scaling properties are identified in the mean extinction times as criticality is approached from above the bifurcation. This noise-induced stabilisation of the delays is also found in the two-dimensional system. |
| first_indexed | 2024-03-12T16:31:36Z |
| format | Article |
| id | doaj.art-839007d4d88d41a3b7b329760d3347ed |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:31:36Z |
| publishDate | 2020-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-839007d4d88d41a3b7b329760d3347ed2023-08-08T15:27:36ZengIOP PublishingNew Journal of Physics1367-26302020-01-0122909306410.1088/1367-2630/abb549Noise-induced stabilization of saddle-node ghostsJosep Sardanyés0https://orcid.org/0000-0001-7225-5158Carles Raich1Tomás Alarcón2https://orcid.org/0000-0002-8566-3676Centre de Recerca Matemàtica, Edifici C, Campus de Bellaterra , 08193 Cerdanyola del Vallès, Barcelona, Spain; Barcelona Graduate School of Mathematics (BGSMath) Edifici C, Campus de Cerdanyola del Vallès , 08193 Bellaterra, Barcelona, SpainDepartament de Matemàtiques i Informàtica (Universitat de Barcelona) , Gran Via de les Corts Catalanes 585, 08007 Barcelona, SpainCentre de Recerca Matemàtica, Edifici C, Campus de Bellaterra , 08193 Cerdanyola del Vallès, Barcelona, Spain; Barcelona Graduate School of Mathematics (BGSMath) Edifici C, Campus de Cerdanyola del Vallès , 08193 Bellaterra, Barcelona, Spain; ICREA , Pg. Lluis Companys 23, 08010 Barcelona, Spain; Departament de Matemàtiques, Universitat Autònoma de Barcelona , Barcelona, SpainIt is known that saddle-node (s-n) bifurcations leave a saddle remnant (or ghost) in the region of the phase space where the annihilation of the fixed points occurred. The corresponding time delay, t _d , found right after the bifurcation is known to follow the scaling law ${t}_{\mathrm{d}}\sim {\left({\epsilon}-{{\epsilon}}_{\mathrm{c}}\right)}^{-1/2}$ , where ϵ and ϵ _c are the control parameter and its critical value, respectively. While the properties of such delays are well understood for deterministic systems, much less is known about how intrinsic noise influences this phenomenon. As a first step towards analysing this issue, in this article we explore a model with autocatalysis and a two-species hypercycle to analyse the impact of noise on delayed transitions in one- and two-dimensional dynamical systems suffering a s-n bifurcation. The first model is investigated with Gillespie simulations and the diffusion approximation, focusing on the behaviour and properties close to the bifurcation. A Fokker–Planck equation is derived, together with the stochastic potential. We show that the slowing down of the dynamics remains robust to noise. In fact, we prove both analytically and numerically that increasing noise lengthens the delays after bifurcation threshold. Furthermore, the inverse square-root scaling law is not robust to fluctuations. By contrast, scaling properties are identified in the mean extinction times as criticality is approached from above the bifurcation. This noise-induced stabilisation of the delays is also found in the two-dimensional system.https://doi.org/10.1088/1367-2630/abb549bifurcationscomplex systemsdelayed transitionsdemographic noisefirst-order phase transitionstransients |
| spellingShingle | Josep Sardanyés Carles Raich Tomás Alarcón Noise-induced stabilization of saddle-node ghosts New Journal of Physics bifurcations complex systems delayed transitions demographic noise first-order phase transitions transients |
| title | Noise-induced stabilization of saddle-node ghosts |
| title_full | Noise-induced stabilization of saddle-node ghosts |
| title_fullStr | Noise-induced stabilization of saddle-node ghosts |
| title_full_unstemmed | Noise-induced stabilization of saddle-node ghosts |
| title_short | Noise-induced stabilization of saddle-node ghosts |
| title_sort | noise induced stabilization of saddle node ghosts |
| topic | bifurcations complex systems delayed transitions demographic noise first-order phase transitions transients |
| url | https://doi.org/10.1088/1367-2630/abb549 |
| work_keys_str_mv | AT josepsardanyes noiseinducedstabilizationofsaddlenodeghosts AT carlesraich noiseinducedstabilizationofsaddlenodeghosts AT tomasalarcon noiseinducedstabilizationofsaddlenodeghosts |