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Organization of Spatially Localized Structures near a Codimension-Three Cusp-Turing Bifurcation.

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Title: Organization of Spatially Localized Structures near a Codimension-Three Cusp-Turing Bifurcation.
Authors: Parra-Rivas, Pedro1 pedro.parra-rivas@uniroma1.it, Champneys, Alan R.2 a.r.champneys@bristol.ac.uk, Al Saadi, Fahad2,3 fa1774@bristol.ac.uk, Gomila, Damia4 damia@ifisc.uib-csic.es, Knobloch, Edgar5 knobloch@berkeley.edu
Source: SIAM Journal on Applied Dynamical Systems. 2023, Vol. 22 Issue 4, p2693-2731. 39p.
Subjects: Lifshitz point (Physics), Bifurcation diagrams, Fluid mechanics, Nonlinear optics, Spatial systems
Abstract: A wide variety of stationary or moving spatially localized structures is present in evolution problems on unbounded domains, governed by higher-than-second-order reversible spatial interactions. This work provides a generic unfolding in one spatial dimension of a certain codimension-three singularity that explains the organization of bifurcation diagrams of such localized states in a variety of contexts, ranging from nonlinear optics to fluid mechanics, mathematical biology, and beyond. The singularity occurs when a cusp bifurcation associated with the onset of bistability between homogeneous steady states encounters a pattern-forming, or Turing, bifurcation. The latter corresponds to a Hamiltonian-Hopf point of the corresponding spatial dynamics problem. Such codimension-three points are sometimes called Lifshitz points in the physics literature. In the simplest case where the spatial system conserves a first integral, the system is described by a canonical fourth-order scalar system. The problem contains three small parameters: two that unfold the cusp bifurcation and one that unfolds the Turing bifurcation. Several cases are revealed, depending on open conditions on the signs of the lowest-order nonlinear terms. Taking the case in which the Turing bifurcation is subcritical, various parameter regimes are considered and the bifurcation diagrams of localized structures are elucidated. A rich bifurcation structure is revealed which involves transitions between regions of localized periodic patterns generated by standard homoclinic snaking, and regions of stationary domains of one homogeneous solution embedded in the other organized in a collapsed snaking structure. The theory is shown to unify previous numerical results obtained in models arising in nonlinear optics, fluid mechanics, and excitable media more generally. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Applied Dynamical Systems is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Organization of Spatially Localized Structures near a Codimension-Three Cusp-Turing Bifurcation.
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  Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Applied+Dynamical+Systems%22">SIAM Journal on Applied Dynamical Systems</searchLink>. 2023, Vol. 22 Issue 4, p2693-2731. 39p.
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  Data: <searchLink fieldCode="DE" term="%22Lifshitz+point+%28Physics%29%22">Lifshitz point (Physics)</searchLink><br /><searchLink fieldCode="DE" term="%22Bifurcation+diagrams%22">Bifurcation diagrams</searchLink><br /><searchLink fieldCode="DE" term="%22Fluid+mechanics%22">Fluid mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+optics%22">Nonlinear optics</searchLink><br /><searchLink fieldCode="DE" term="%22Spatial+systems%22">Spatial systems</searchLink>
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  Data: A wide variety of stationary or moving spatially localized structures is present in evolution problems on unbounded domains, governed by higher-than-second-order reversible spatial interactions. This work provides a generic unfolding in one spatial dimension of a certain codimension-three singularity that explains the organization of bifurcation diagrams of such localized states in a variety of contexts, ranging from nonlinear optics to fluid mechanics, mathematical biology, and beyond. The singularity occurs when a cusp bifurcation associated with the onset of bistability between homogeneous steady states encounters a pattern-forming, or Turing, bifurcation. The latter corresponds to a Hamiltonian-Hopf point of the corresponding spatial dynamics problem. Such codimension-three points are sometimes called Lifshitz points in the physics literature. In the simplest case where the spatial system conserves a first integral, the system is described by a canonical fourth-order scalar system. The problem contains three small parameters: two that unfold the cusp bifurcation and one that unfolds the Turing bifurcation. Several cases are revealed, depending on open conditions on the signs of the lowest-order nonlinear terms. Taking the case in which the Turing bifurcation is subcritical, various parameter regimes are considered and the bifurcation diagrams of localized structures are elucidated. A rich bifurcation structure is revealed which involves transitions between regions of localized periodic patterns generated by standard homoclinic snaking, and regions of stationary domains of one homogeneous solution embedded in the other organized in a collapsed snaking structure. The theory is shown to unify previous numerical results obtained in models arising in nonlinear optics, fluid mechanics, and excitable media more generally. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of SIAM Journal on Applied Dynamical Systems is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/22M1514234
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      – SubjectFull: Nonlinear optics
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      – TitleFull: Organization of Spatially Localized Structures near a Codimension-Three Cusp-Turing Bifurcation.
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              Text: 2023
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