{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T02:46:11Z","timestamp":1774579571992,"version":"3.50.1"},"reference-count":104,"publisher":"AIP Publishing","issue":"8","license":[{"start":{"date-parts":[[2023,8,10]],"date-time":"2023-08-10T00:00:00Z","timestamp":1691625600000},"content-version":"vor","delay-in-days":9,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2023,8,10]],"date-time":"2023-08-10T00:00:00Z","timestamp":1691625600000},"content-version":"tdm","delay-in-days":9,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 910"],"award-info":[{"award-number":["SFB 910"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["Project A4"],"award-info":[{"award-number":["Project A4"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04459\/2020"],"award-info":[{"award-number":["UIDB\/04459\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2023,8,1]]},"abstract":"<jats:p>We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) ut=uxx+f(x,u,ux) on the unit interval 0&amp;lt;x&amp;lt;1, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions ut=0. Specifically, we address meanders with only three \u201cnoses,\u201d each of which is innermost to a nested family of upper or lower meander arcs. The Chafee\u2013Infante paradigm, with cubic nonlinearity f=f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v1\u21ddv2 between equilibrium vertices v1,v2 of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold Wu(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d\u22121)-sphere boundary of the global attractor.<\/jats:p>","DOI":"10.1063\/5.0147634","type":"journal-article","created":{"date-parts":[[2023,8,10]],"date-time":"2023-08-10T13:45:57Z","timestamp":1691675157000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["Design of Sturm global attractors 1: Meanders with three noses, and reversibility"],"prefix":"10.1063","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1141-8475","authenticated-orcid":false,"given":"Bernold","family":"Fiedler","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mathematik, Freie Universit\u00e4t Berlin 1 , Arnimallee 3, 14195 Berlin, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3994-2834","authenticated-orcid":false,"given":"Carlos","family":"Rocha","sequence":"additional","affiliation":[{"name":"Instituto Superior T\u00e9cnico 2 , Avenida Rovisco Pais, 1049-001 Lisboa, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2023,8,10]]},"reference":[{"key":"2023081013455073700_c1","doi-asserted-by":"publisher","first-page":"1085","DOI":"10.1016\/0001-6160(79)90196-2","article-title":"A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening","volume":"27","year":"1979","journal-title":"Acta Metall."},{"key":"2023081013455073700_c2","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1016\/0022-0396(86)90093-8","article-title":"The Morse-Smale property for a semi-linear parabolic equation","volume":"62","year":"1986","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c3","first-page":"79","article-title":"The zero set of a solution of a parabolic equation","volume":"390","year":"1988","journal-title":"J. Reine Angew. Math."},{"key":"2023081013455073700_c4","doi-asserted-by":"publisher","first-page":"601","DOI":"10.4310\/jdg\/1214446558","article-title":"On the formation of singularities in the curve shortening flow","volume":"33","year":"1991","journal-title":"J. Differ. Geom."},{"key":"2023081013455073700_c5","doi-asserted-by":"publisher","first-page":"212","DOI":"10.1016\/0022-0396(87)90147-1","article-title":"Stable transition layers in a semilinear boundary value problem","volume":"67","year":"1987","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c6","volume-title":"The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts","year":"1975"},{"key":"2023081013455073700_c7","volume-title":"Attractors of Evolution Equations","year":"1992"},{"key":"2023081013455073700_c8","unstructured":"N. Ben-Gal , \u201cGrow-up solutions and heteroclinics to infinity for scalar parabolic PDEs,\u201d Ph.D. thesis, Div. Appl. Math. (Brown University, 2010)."},{"key":"2023081013455073700_c9"},{"key":"2023081013455073700_c10","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/BF02698544","article-title":"Morse theory indomitable","volume":"68","year":"1988","journal-title":"Publ. Math. I.H.\u00c9.S."},{"key":"2023081013455073700_c11","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1090\/memo\/0285","article-title":"Convergence of solutions of the Kolmogorov equation to travelling waves","volume":"44","year":"1983","journal-title":"Mem. Am. Math. Soc."},{"key":"2023081013455073700_c12","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/978-3-322-96656-8_2","article-title":"Connecting orbits in scalar reaction diffusion equations","volume":"1","year":"1988","journal-title":"Dyn. Rep."},{"key":"2023081013455073700_c13","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1016\/0022-0396(89)90180-0","article-title":"Connecting orbits in scalar reaction diffusion equations II: The complete solution","volume":"81","year":"1989","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c14","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1017\/prm.2018.51","article-title":"Autonomous and non-autonomous unbounded attractors under perturbations","volume":"149","year":"2019","journal-title":"Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci."},{"key":"2023081013455073700_c15","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1080\/00036817408839081","article-title":"A bifurcation problem for a nonlinear parabolic equation","volume":"4","year":"1974","journal-title":"J. Appl. Anal."},{"key":"2023081013455073700_c16","volume-title":"Attractors for Equations of Mathematical Physics","year":"2002"},{"key":"2023081013455073700_c17","doi-asserted-by":"publisher","first-page":"1004","DOI":"10.1137\/19M1300145","article-title":"Ginzburg-Landau spiral waves in circular and spherical geometries","volume":"53","year":"2021","journal-title":"SIAM J. Math. Anal."},{"key":"2023081013455073700_c18","doi-asserted-by":"publisher","first-page":"1959","DOI":"10.1137\/20M1378739","article-title":"Ginzburg-Landau patterns in circular and spherical geometries: Vortices, spirals, and attractors","volume":"20","year":"2021","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"2023081013455073700_c19","unstructured":"V. Delecroix , \u201cAsymptotics of lieanders with fixed composition sizes,\u201d arXiv:1812.03912 (2018)."},{"key":"2023081013455073700_c20","doi-asserted-by":"publisher","first-page":"e4","DOI":"10.1017\/fmp.2020.2","article-title":"Enumeration of meanders and Masur-Veech volumes","volume":"8","year":"2020","journal-title":"Forum Math., Pi"},{"key":"2023081013455073700_c21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01443322","article-title":"Gruppentheoretische studien","volume":"20","year":"1882","journal-title":"Math. Ann."},{"key":"2023081013455073700_c22","volume-title":"Exponential Attractors for Dissipative Evolution Equations","year":"1994"},{"key":"2023081013455073700_c23","first-page":"67","article-title":"Global attractors of one-dimensional parabolic equations: Sixteen examples","volume":"4","year":"1994","journal-title":"Tatra Mountains Math. Publ."},{"key":"2023081013455073700_c24"},{"key":"2023081013455073700_c25","doi-asserted-by":"publisher","first-page":"3691","DOI":"10.1142\/S0218127405014325","article-title":"Roots and centralizers of Anosov diffeomorphisms on tori","volume":"15","year":"2005","journal-title":"Int. J. Bifurcation Chaos"},{"key":"2023081013455073700_c26","doi-asserted-by":"publisher","first-page":"177","DOI":"10.3934\/dcdss.2020344","article-title":"Global Hopf bifurcation in networks with fast feedback cycles","volume":"14","year":"2021","journal-title":"Discrete Contin. Dyn. Syst. S"},{"key":"2023081013455073700_c27","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1007\/s000330050151","article-title":"Basins of attraction in strongly damped coupled mechanical oscillators: A global example","volume":"50","year":"1999","journal-title":"Z. Angew. Math. Phys."},{"key":"2023081013455073700_c28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.11606\/issn.2316-9028.v6i2p247-275","article-title":"Rainbow meanders and Cartesian billiards","volume":"6","year":"2013","journal-title":"S\u00e3o Paulo J. Math. Sci."},{"key":"2023081013455073700_c29","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1137\/S0036141097316147","article-title":"A Lyapunov function for tridiagonal competitive-cooperative systems","volume":"30","year":"1999","journal-title":"SIAM J. Math. Anal."},{"key":"2023081013455073700_c30","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1070\/RM2014v069n03ABEH004897","article-title":"An explicit Lyapunov function for reflection symmetric parabolic differential equations on the circle","volume":"69","year":"2014","journal-title":"Russ. Math. Surv."},{"key":"2023081013455073700_c31","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1016\/j.jde.2004.02.012","article-title":"Multiplicity of rotating spirals under curvature flows with normal tip motion","volume":"205","year":"2004","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c32","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s10231-004-0145-1","article-title":"Rotating spirals of curvature flows: A center manifold approach","volume":"185","year":"2006","journal-title":"Ann. Mat. Pura Appl."},{"key":"2023081013455073700_c33","doi-asserted-by":"publisher","first-page":"867","DOI":"10.1007\/s10884-007-9083-0","article-title":"Blow-up shapes on fast unstable manifolds of one-dimensional reaction-diffusion equations","volume":"19","year":"2007","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c34","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1006\/jdeq.1996.0031","article-title":"Heteroclinic orbits of semilinear parabolic equations","volume":"125","year":"1996","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c35","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1006\/jdeq.1998.3532","article-title":"Realization of meander permutations by boundary value problems","volume":"156","year":"1999","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c36","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1090\/S0002-9947-99-02209-6","article-title":"Orbit equivalence of global attractors of semilinear parabolic differential equations","volume":"352","year":"2000","journal-title":"Trans. Am. Math. Soc."},{"key":"2023081013455073700_c37","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1515\/CRELLE.2009.076","article-title":"Connectivity and design of planar global attractors of Sturm type. I: Bipolar orientations and Hamiltonian paths","volume":"635","year":"2009","journal-title":"Crelle J. Reine Angew. Math."},{"key":"2023081013455073700_c38","doi-asserted-by":"publisher","first-page":"1255","DOI":"10.1016\/j.jde.2007.09.015","article-title":"Connectivity and design of planar global attractors of Sturm type. II: Connection graphs","volume":"244","year":"2008","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c39","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s10884-009-9149-2","article-title":"Connectivity and design of planar global attractors of Sturm type. III: Small and Platonic examples","volume":"22","year":"2009","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c40","doi-asserted-by":"publisher","first-page":"5099","DOI":"10.3934\/dcds.2014.34.5099","article-title":"Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes","volume":"34","year":"2014","journal-title":"Discrete Contin. Dyn. Sys."},{"key":"2023081013455073700_c41","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1007\/s10884-013-9311-8","article-title":"Schoenflies spheres as boundaries of bounded unstable manifolds in gradient Sturm systems","volume":"27","year":"2015","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c42","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1007\/s40863-017-0082-8","article-title":"Sturm 3-balls and global attractors 1: Thom-Smale complexes and meanders","volume":"12","year":"2018","journal-title":"S\u00e3o Paulo J. Math. Sci."},{"key":"2023081013455073700_c43","doi-asserted-by":"publisher","first-page":"1549","DOI":"10.1007\/s10884-018-9665-z","article-title":"Sturm 3-balls and global attractors 2: Design of Thom-Smale complexes","volume":"31","year":"2018","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c44","doi-asserted-by":"publisher","first-page":"3479","DOI":"10.3934\/dcds.2018149","article-title":"Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes","volume":"38","year":"2018","journal-title":"Discrete Contin. Dyn. Syst. A"},{"key":"2023081013455073700_c45","doi-asserted-by":"publisher","first-page":"2787","DOI":"10.1007\/s10884-020-09836-5","article-title":"Boundary orders and geometry of the signed Thom-Smale complex for Sturm global attractors","volume":"34","year":"2020","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c46","doi-asserted-by":"crossref","unstructured":"B. Fiedler and C.Rocha, \u201cDesign of Sturm global attractors 2: Time-reversible Chafee-Infante lattices of 3-nose meanders,\u201d arXiv:2306.05232 (2023).","DOI":"10.1007\/s40863-023-00385-5"},{"key":"2023081013455073700_c47","first-page":"151","article-title":"Dynamics of piecewise-autonomous bistable parabolic equations","volume":"31","year":"2002","journal-title":"Fields Inst. Commun."},{"key":"2023081013455073700_c48","doi-asserted-by":"publisher","first-page":"588","DOI":"10.1016\/j.jde.2011.08.013","article-title":"A permutation characterization of Sturm global attractors of Hamilton type","volume":"252","year":"2012","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c49","doi-asserted-by":"publisher","first-page":"617","DOI":"10.3934\/nhm.2012.7.617","article-title":"Sturm global attractors for S1-equivariant parabolic equations","volume":"7","year":"2012","journal-title":"Networks Het. Media"},{"key":"2023081013455073700_c50"},{"key":"2023081013455073700_c51","doi-asserted-by":"publisher","first-page":"879","DOI":"10.1016\/S0021-7824(01)80002-7","article-title":"Large patterns of elliptic systems in infinite cylinders","volume":"77","year":"1998","journal-title":"J. Math. Pures Appl."},{"key":"2023081013455073700_c52"},{"key":"2023081013455073700_c53","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1111\/j.1469-1809.1937.tb02153.x","article-title":"The wave of advance of advantageous genes","volume":"7","year":"1937","journal-title":"Ann. Eugenics"},{"key":"2023081013455073700_c54","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/0040-9383(79)90003-X","article-title":"Morse-Smale flows and homotopy theory","volume":"18","year":"1979","journal-title":"Topology"},{"key":"2023081013455073700_c55","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1017\/S0308210500027748","article-title":"Jacobi matrices and transversality","volume":"109","year":"1988","journal-title":"Proc. R. Soc. Edinburgh, Sect. A"},{"key":"2023081013455073700_c56","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/0022-0396(91)90134-U","article-title":"A permutation related to the dynamics of a scalar parabolic PDE","volume":"91","year":"1991","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c57","volume-title":"Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications","year":"2004"},{"key":"2023081013455073700_c58"},{"key":"2023081013455073700_c59"},{"key":"2023081013455073700_c60","volume-title":"Dynamics in Infinite Dimensions","year":"2002"},{"key":"2023081013455073700_c61","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1017\/S0308210500021491","article-title":"Heteroclinic orbits between rotating waves in hyperbolic balance laws","volume":"129","year":"1999","journal-title":"Proc. R. Soc. Edinburgh, Sect. A"},{"key":"2023081013455073700_c62","doi-asserted-by":"publisher","first-page":"531","DOI":"10.3934\/dcds.2005.12.531","article-title":"Describing a class of global attractors via symbol sequences","volume":"12","year":"2005","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"2023081013455073700_c63","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1016\/0022-0396(91)90096-R","article-title":"A dynamical system approach to a phase transition problem","volume":"94","year":"1991","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c64"},{"key":"2023081013455073700_c65","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/0022-0396(85)90153-6","article-title":"Some infinite-dimensional Morse-Smale systems defined by parabolic differential equations","volume":"59","year":"1985","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c66"},{"key":"2023081013455073700_c67","volume-title":"Meanders","year":"2017"},{"key":"2023081013455073700_c68","first-page":"1","article-title":"Study of a diffusion equation that is related to the growth of a quality of matter and its application to a biological problem","volume":"1","year":"1937","journal-title":"Moscow Univ. Math. Bull."},{"key":"2023081013455073700_c69","volume-title":"Attractors for Semigroups and Evolution Equations","year":"1991"},{"key":"2023081013455073700_c70","doi-asserted-by":"publisher","first-page":"4642","DOI":"10.1016\/j.jde.2018.06.018","article-title":"Sturm attractors for quasilinear parabolic equations","volume":"265","year":"2018","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c71","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1007\/s10884-018-9720-9","article-title":"Sturm attractors for quasilinear parabolic equations with singular coefficients","volume":"32","year":"2020","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c72","doi-asserted-by":"publisher","DOI":"10.1007\/s13163-022-00435-0","article-title":"Sturm attractors for fully nonlinear parabolic equations","year":"2022","journal-title":"Rev. Mat. Complut."},{"key":"2023081013455073700_c73","doi-asserted-by":"crossref","unstructured":"Ph. Lappicy and EsterBeatriz, \u201cAn energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension,\u201d arXiv:2201.04215 (2022).","DOI":"10.1007\/s13163-022-00435-0"},{"key":"2023081013455073700_c74","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s40863-018-00115-2","article-title":"A Lyapunov function for fully nonlinear parabolic equations in one spatial variable","volume":"13","year":"2019","journal-title":"S\u00e0o Paulo J. Math. Sci."},{"key":"2023081013455073700_c75","doi-asserted-by":"publisher","first-page":"313","DOI":"10.4171\/PM\/2021","article-title":"Slowly non-dissipative equations with oscillating growth","volume":"75","year":"2018","journal-title":"Port. Math. (N.S.)"},{"key":"2023081013455073700_c76","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1016\/0022-0396(88)90157-X","article-title":"Morse decompositions for delay-differential equations","volume":"72","year":"1988","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c77","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1006\/jdeq.1996.0037","article-title":"The Poincar\u00e9-Bendixson theorem for monotone cyclic feedback systems with delay","volume":"125","year":"1996","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c78","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/BF01054041","article-title":"The Poincar\u00e9-Bendixson theorem for monotone cyclic feedback systems","volume":"2","year":"1990","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c79","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1215\/kjm\/1250522572","article-title":"Convergence of solutions of one-dimensional semilinear parabolic equations","volume":"18","year":"1978","journal-title":"J. Math. Kyoto Univ."},{"key":"2023081013455073700_c80","first-page":"401","article-title":"Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation","volume":"29","year":"1982","journal-title":"J. Fac. Sci. Univ. Tokyo Sec. IA"},{"key":"2023081013455073700_c81","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3934\/dcds.1997.3.1","article-title":"The global attractor of semilinear parabolic equations on S1","volume":"3","year":"1997","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"2023081013455073700_c82","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1006\/jdeq.1994.1070","article-title":"Essential manifolds for elliptic problems in infinite cylinders","volume":"110","year":"1994","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c83","doi-asserted-by":"publisher","first-page":"1199","DOI":"10.1137\/S0036141093250827","article-title":"Global asymptotic dynamics of gradient-like bistable equations","volume":"26","year":"1995","journal-title":"SIAM J. Math. Anal."},{"key":"2023081013455073700_c84","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1016\/0022-0396(89)90065-X","article-title":"The singular limit dynamics of semilinear damped wave equations","volume":"78","year":"1989","journal-title":"J. Differ. Equ."},{"key":"2023081013455073700_c85"},{"key":"2023081013455073700_c86"},{"key":"2023081013455073700_c87","volume-title":"Geometric Theory of Dynamical Systems. An Introduction","year":"1982"},{"key":"2023081013455073700_c88","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1090\/pspum\/014\/0267603","article-title":"Structural stability theorems","volume":"14","year":"1970","journal-title":"Global Anal. Proc. Symp. Pure Math."},{"key":"2023081013455073700_c89","volume-title":"Semigroups of Linear Operators and Applications to Partial Differential Equations","year":"1983"},{"key":"2023081013455073700_c90","doi-asserted-by":"publisher","first-page":"3860","DOI":"10.1137\/15M1051476","article-title":"Unbounded Sturm global attractors for semilinear parabolic equations on the circle","volume":"48","year":"2016","journal-title":"SIAM J. Math. Anal."},{"key":"2023081013455073700_c91","volume-title":"Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States","year":"2007"},{"key":"2023081013455073700_c92"},{"key":"2023081013455073700_c93","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1007\/BF01049100","article-title":"Properties of the attractor of a scalar parabolic PDE","volume":"3","year":"1991","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c94","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-021-10053-x","article-title":"Meanders, zero numbers and the cell structure of Sturm global attractors","year":"2021","journal-title":"J. Dyn. Diff. Eqn."},{"key":"2023081013455073700_c95","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1007\/BF02218843","article-title":"Existence of fast traveling waves for some parabolic equations: A dynamical systems approach","volume":"8","year":"1996","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c96","volume-title":"Dynamics of Evolutionary Equations","year":"2002"},{"key":"2023081013455073700_c97","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/BF00276388","article-title":"Oscillations and multiple steady states in a cyclic gene model with repression","volume":"25","year":"1987","journal-title":"J. Math. Biol."},{"key":"2023081013455073700_c98","first-page":"373","article-title":"Sur une classe d\u2019\u00e9quations \u00e0 diff\u00e9rences partielles","volume":"1","year":"1836","journal-title":"J. Math. Pure Appl."},{"key":"2023081013455073700_c99","volume-title":"Equations of Evolution","year":"1979"},{"key":"2023081013455073700_c100","volume-title":"Infinite-Dimensional Dynamical Systems in Mechanics and Physics","year":"1988"},{"key":"2023081013455073700_c101","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1023\/A:1012967428328","article-title":"Geometry of heteroclinic cascades in scalar parabolic differential equations","volume":"14","year":"2002","journal-title":"J. Dyn. Differ. Equ."},{"key":"2023081013455073700_c102","first-page":"203"},{"key":"2023081013455073700_c103","first-page":"17","article-title":"Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable","volume":"4","year":"1968","journal-title":"Differ. Equ."},{"key":"2023081013455073700_c104"}],"container-title":["Chaos: An Interdisciplinary Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0147634\/18081677\/083127_1_5.0147634.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0147634\/18081677\/083127_1_5.0147634.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0147634\/18081677\/083127_1_5.0147634.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T18:54:37Z","timestamp":1746644077000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/33\/8\/083127\/2906436\/Design-of-Sturm-global-attractors-1-Meanders-with"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,1]]},"references-count":104,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2023,8,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0147634","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,8]]},"published":{"date-parts":[[2023,8,1]]},"article-number":"083127"}}