Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system
A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary front...
| Main Authors: | , , , |
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| Format: | Article |
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Royal Society
2017
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| Online Access: | https://eprints.nottingham.ac.uk/47195/ |
| _version_ | 1848797486766882816 |
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| author | Avitabile, Daniele Desroches, Mathieu Knobloch, Edgar Krupa, Martin |
| author_facet | Avitabile, Daniele Desroches, Mathieu Knobloch, Edgar Krupa, Martin |
| author_sort | Avitabile, Daniele |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for nonzero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary condition. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial-dynamical system. |
| first_indexed | 2025-11-14T20:04:39Z |
| format | Article |
| id | nottingham-47195 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:04:39Z |
| publishDate | 2017 |
| publisher | Royal Society |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-471952020-05-04T19:16:30Z https://eprints.nottingham.ac.uk/47195/ Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system Avitabile, Daniele Desroches, Mathieu Knobloch, Edgar Krupa, Martin A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for nonzero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary condition. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial-dynamical system. Royal Society 2017-11-08 Article PeerReviewed Avitabile, Daniele, Desroches, Mathieu, Knobloch, Edgar and Krupa, Martin (2017) Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473 (2207). ISSN 1471-2946 http://rspa.royalsocietypublishing.org/content/473/2207/20170018 doi:10.1098/rspa.2017.0018 doi:10.1098/rspa.2017.0018 |
| spellingShingle | Avitabile, Daniele Desroches, Mathieu Knobloch, Edgar Krupa, Martin Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system |
| title | Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system |
| title_full | Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system |
| title_fullStr | Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system |
| title_full_unstemmed | Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system |
| title_short | Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system |
| title_sort | ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system |
| url | https://eprints.nottingham.ac.uk/47195/ https://eprints.nottingham.ac.uk/47195/ https://eprints.nottingham.ac.uk/47195/ |