{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,6]],"date-time":"2024-09-06T08:49:22Z","timestamp":1725612562385},"publisher-location":"New York, NY","reference-count":52,"publisher":"Springer New York","isbn-type":[{"type":"print","value":"9781461271925"},{"type":"electronic","value":"9781461215585"}],"license":[{"start":{"date-parts":[[1999,1,1]],"date-time":"1999-01-01T00:00:00Z","timestamp":915148800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1999]]},"DOI":"10.1007\/978-1-4612-1558-5_7","type":"book-chapter","created":{"date-parts":[[2011,9,22]],"date-time":"2011-09-22T21:51:37Z","timestamp":1316728297000},"page":"83-99","source":"Crossref","is-referenced-by-count":5,"title":["Spatial Hidden Symmetries in Pattern Formation"],"prefix":"10.1007","author":[{"given":"M. Gabriela M.","family":"Gomes","sequence":"first","affiliation":[]},{"given":"Isabel S.","family":"Labouriau","sequence":"additional","affiliation":[]},{"given":"Eliana M.","family":"Pinho","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"7_CR1","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1017\/S0305004100066500","volume":"101","author":"D Armbruster","year":"1987","unstructured":"Armbruster, D. & Dangelmayr, G. (1987) Coupled stationary bifurcations in non-flux boundary value problems. Math. Proc. Camb. Phil. Soc.\n                101, 167\u2013192.","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"7_CR2","unstructured":"Ashwin, P. (1992) High corank steady-state mode interactions on a rectangle, in Bifurcation and Symmetry (ed. E.L. Algower, K. Borner & M. Golubitsky) ISNM vol. 104, Basel, Birkh\u00e4user, 23\u201333."},{"key":"7_CR3","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1007\/BF02218859","volume":"6","author":"P Ashwin","year":"1994","unstructured":"Ashwin, P. & Mei, Z. (1994) A Hopf bifurcation with Robin boundary conditions, Journal of Dynamics and Differential Equations\n                6, 487\u2013503.","journal-title":"Journal of Dynamics and Differential Equations"},{"key":"7_CR4","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1088\/0951-7715\/8\/5\/004","volume":"8","author":"P Ashwin","year":"1995","unstructured":"Ashwin, P. & Mei, Z. (1995) Normal form for Hopf bifurcation of partial differential equations on the square, Nonlinearity\n                8, 715\u2013734.","journal-title":"Nonlinearity"},{"key":"7_CR5","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1016\/0167-2789(94)00208-8","volume":"81","author":"P. Ashwin","year":"1995","unstructured":"Ashwin, P., Moroz, I. & Roberts, M. (1995) Bifurcations of stationary, standing, and travelling waves in triply diffusive convection. Physica D 81, 374\u2013397.","journal-title":"Physica D"},{"key":"7_CR6","unstructured":"Castro, S.B.S.D. (1990) Symmetry and bifurcation of periodic solutions in Neumann boundary value problems. MSc Thesis, University of Warwick."},{"key":"7_CR7","doi-asserted-by":"crossref","unstructured":"Chien, C.S., Kuo, Y.J. & Mei, Z. (1998) Symmetry and scaling properties of the von K\u00e1rm\u00e1n equations. To appear in ZAMM.","DOI":"10.1007\/PL00001487"},{"key":"7_CR8","first-page":"429","volume":"52D","author":"JD Crawford","year":"1991","unstructured":"Crawford, J.D. (1991) Normal forms for driven surface waves: boundary conditions, symmetry and genericity. Physica\n                52D, 429\u2013457.","journal-title":"Physica"},{"key":"7_CR9","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1103\/PhysRevLett.67.441","volume":"67","author":"JD Crawford","year":"1991","unstructured":"Crawford, J.D. (1991) Surface waves in non-square containers with square symmetry. Phys. Rev. Lett.\n                67, 441\u2013444.","journal-title":"Phys. Rev. Lett."},{"key":"7_CR10","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1088\/0951-7715\/7\/3\/002","volume":"7","author":"JD Crawford","year":"1994","unstructured":"Crawford, J.D. (1994) D\n                4+T\n                2 mode interactions and hidden rotation symmetry. Nonlinearity\n                7, 697\u2013739.","journal-title":"Nonlinearity"},{"key":"7_CR11","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1088\/0951-7715\/6\/2\/001","volume":"6","author":"JD Crawford","year":"1993","unstructured":"Crawford, J.D., Gollub, J.P. & Lane, D. (1993) Hidden symmetries of parametrically forced waves. Nonlinearity\n                6, 119\u2013164.","journal-title":"Nonlinearity"},{"key":"7_CR12","volume-title":"Singularity Theory and Its Applications, Warwick 1989","author":"JD Crawford","year":"1991","unstructured":"Crawford, J.D., Golubitsky, M., Gomes, M.G.M., Knobloch, E. & Stewart, I.N. (1991) Boundary conditions as symmetry constraints, in Singularity Theory and Its Applications, Warwick 1989 (ed. R.M. Roberts & I.N. Stewart) Lecture Notes in Mathematics, vol 2, Heidelberg, Springer."},{"key":"7_CR13","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1103\/RevModPhys.65.851","volume":"65","author":"MC Cross","year":"1993","unstructured":"Cross, M.C. & Hohenberg, P.C. (1993) Pattern formation outside of equilibrium. Rev. Mod. Phys.\n                65, 851\u20131112.","journal-title":"Rev. Mod. Phys."},{"key":"7_CR14","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1090\/conm\/056\/855084","volume-title":"Multiparameter Bifurcation Theory","author":"G Dangelmayr","year":"1986","unstructured":"Dangelmayr, G. & Armbruster, D. (1986) Steady-state mode interactions in the presence of O(2) symmetry and in non-flux boundary value problems, in Multiparameter Bifurcation Theory (ed. M. Golubitsky & J. Guckenheimer) Contemporary Math. 56, Providence, RI, American Mathematical Society, 53\u201368."},{"key":"7_CR15","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1007\/BF00944740","volume":"43","author":"B Dionne","year":"1992","unstructured":"Dionne, B. & Golubitsky, M. (1992) Planforms in Two and Three Dimensions. Z. Angew. Math. Phys.\n                43, 36\u201362.","journal-title":"Z. Angew. Math. Phys."},{"issue":"4","key":"7_CR16","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1007\/BF00948482","volume":"44","author":"B Dionne","year":"1993","unstructured":"Dionne, B. (1993) Planforms in Three Dimensions. Z. Angew. Math. Phys.\n                44(4), 673\u2013694.","journal-title":"Z. Angew. Math. Phys."},{"issue":"1","key":"7_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1063\/1.165974","volume":"3","author":"IR Epstein","year":"1993","unstructured":"Epstein, I.R. & Golubitsky, M. (1993) Symmetric patterns in linear arrays of coupled celld. Chaos\n                3(1), 1\u20135.","journal-title":"Chaos"},{"key":"7_CR18","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/BF01209066","volume":"1","author":"M Field","year":"1990","unstructured":"Field, M., Golubitsky, M. & Stewart, I.N. (1990) Bifurcations on hemispheres. J. Nonlinear Sci.\n                1, 201\u2013223.","journal-title":"J. Nonlinear Sci."},{"key":"7_CR19","first-page":"1","volume":"5D","author":"H Fujii","year":"1982","unstructured":"Fujii, H., Mimura, M. & Nishiura, Y. (1982) A picture of the global bifurcation diagram in ecological interacting and diffusing systems. Physica\n                5D, 1\u201342.","journal-title":"Physica"},{"key":"7_CR20","first-page":"487","volume":"208","author":"D Gillis","year":"1997","unstructured":"Gillis, D. & Golubitsky, M. (1997) Pattern in square arrays of coupled cells. JMAA\n                208, 487\u2013509.","journal-title":"JMAA"},{"key":"7_CR21","volume-title":"Appl. Math. Sci.","author":"M Golubitsky","year":"1985","unstructured":"Golubitsky, M. & Schaeffer, D. (1985) Singularities and Groups in Bifurcation Theory. Vol.1, Appl. Math. Sci. 51, Springer-Verlag, New York."},{"key":"7_CR22","doi-asserted-by":"crossref","unstructured":"Golubitsky, M., Stewart, I.N. & Schaeffer, D. (1988) Singularities and Groups in Bifurcation Theory. Vol.2, Appl. Math. Sci. 69, Springer-Verlag, New York.","DOI":"10.1007\/978-1-4612-4574-2"},{"key":"7_CR23","unstructured":"Gomes, M.G.M. (1989) Steady-state mode interactions in rectangular domains. MSc Thesis, University of Warwick."},{"key":"7_CR24","unstructured":"Gomes, M.G.M. (1992) Symmetries in bifurcation theory: the appropriate context PhD Thesis, University of Warwick."},{"key":"7_CR25","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1088\/0951-7715\/7\/1\/012","volume":"7","author":"MGM Gomes","year":"1994","unstructured":"Gomes, M.G.M. & Stewart, I.N. (1994) Steady PDEs on generalized rectangles: a change of genericity in mode interactions. Nonlinearity 7 (1994) 253\u2013272.","journal-title":"Nonlinearity"},{"key":"7_CR26","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/978-94-011-0956-7_13","volume-title":"Dynamics, Bifurcations, and Symmetries: New Trends and New Tools, Carg\u00e8se 1993","author":"MGM Gomes","year":"1994","unstructured":"Gomes, M.G.M. & Stewart, I.N. (1994) Hopf bifurcations on generalized rectangles with Neumann boundary conditions, in Dynamics, Bifurcations, and Symmetries: New Trends and New Tools, Carg\u00e8se 1993, (ed. P Chossat), Kluwer Academic Publishers, 139\u2013158."},{"issue":"1","key":"7_CR27","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1142\/S0218127497000108","volume":"7","author":"MGM Gomes","year":"1997","unstructured":"Gomes, M.G.M. & Stewart, I.N. (1997) Symmetry of Generic Bifurcations in Cubic Domains. Int. Jour. Bif. and Chaos, 7(1) (1997) 147\u2013171.","journal-title":"Int. Jour. Bif. and Chaos"},{"key":"7_CR28","volume-title":"Appl. Math. Sci.","author":"J Guckenheimer","year":"1983","unstructured":"Guckenheimer, J. & Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci. 42, Springer-Verlag, New York."},{"key":"7_CR29","doi-asserted-by":"crossref","unstructured":"Complex Spatial Patterns on Planar Continua. Phys. Rev. Lett.\n                71(9), 1367\u20131370.","DOI":"10.1103\/PhysRevLett.71.1367"},{"key":"7_CR30","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/BF03323073","volume":"21","author":"TJ Healey","year":"1992","unstructured":"Healey, T.J. & Kielh\u00f6fer (1992) Hidden symmetry of fully nonlinear boundary conditions in elliptic equations: global bifurcation and nodal structure. Results in Math.\n                21, 83\u201392.","journal-title":"Results in Math."},{"issue":"6","key":"7_CR31","doi-asserted-by":"publisher","first-page":"537","DOI":"10.1007\/s003329900039","volume":"7","author":"P Hirschberg","year":"1997","unstructured":"Hirschberg, P. & Knobloch, E. (1997) Mode Interactions in Large Aspect Ratio Convection. J. Nonlinear Science, 7(6), 537\u2013556.","journal-title":"J. Nonlinear Science"},{"key":"7_CR32","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1137\/0153006","volume":"53","author":"MD Impey","year":"1993","unstructured":"Impey, M.D., Riley, D.S. & Wheeler, A.A. (1993) Bifurcation analysis of cellular interfaces in unidirectional solidification of a dilute binary mixture. SIAM J. Appl. Math.\n                53, 78\u201395.","journal-title":"SIAM J. Appl. Math."},{"issue":"3","key":"7_CR33","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1080\/02681119608806223","volume":"11","author":"M. Impey","year":"1996","unstructured":"Impey, M., Roberts, M. & Stewart. I. (1996) Hidden Symmetries in Lapwood Convection. Dyn. Stab. Sys., 11 (3), 155\u2013192.","journal-title":"Dyn. Stab. Sys."},{"key":"7_CR34","first-page":"229","volume":"8","author":"G Iooss","year":"1989","unstructured":"Iooss, G., Mielke, A. & Demay, Y. (1989) Theory of the steady Ginzburg-Landau equation in hydrodynamic stability problems. Eur. J. Mech. B\/Fluids\n                8, 229.","journal-title":"Eur. J. Mech. B\/Fluids"},{"key":"7_CR35","unstructured":"Manoel, M. (1995) Hidden Symmetries in Bifurcation Problems: the Singularity Theory. PhD Thesis, University of Warwick."},{"key":"7_CR36","unstructured":"Manoel, M. & Stewart, I. (1995) On the Classification of Bifurcations with Hidden Symmetries. University of Warwick preprint."},{"key":"7_CR37","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/978-3-0348-7536-3_23","volume-title":"Bifurcation and symmetry: Cross Influence and Application","author":"Z Mei","year":"1992","unstructured":"Mei, Z. (1992) Utilization of scaling laws and symmetries in the path following of a semilinear elliptic problem, in Bifurcation and symmetry: Cross Influence and Application (eds. E.L. Allgower, K. B\u00f6hmer and M. Golubitsky), Birkh\u00e4user Verlag, Basel, 263\u2013273."},{"key":"7_CR38","first-page":"479","volume":"123A","author":"Z Mei","year":"1993","unstructured":"Mei, Z. (1993) Solution branches of a semilinear elliptic problem at corank 2 bifurcation points with Neumann boundary conditions, Proceedings of the Royal Society of Edinburgh\n                123A, 479\u2013495.","journal-title":"Proceedings of the Royal Society of Edinburgh"},{"key":"7_CR39","volume-title":"Models of Biological Pattern Formation","author":"H Meinhardt","year":"1982","unstructured":"Meinhardt, H. (1982) Models of Biological Pattern Formation. Academic Press, New York."},{"key":"7_CR40","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s003329900041","volume":"8","author":"I Melbourne","year":"1998","unstructured":"Melbourne, I. (1998) Derivation of the time-dependent Ginzburg-Landau equation on the line. J. Nonlin. Sci.\n                8, 1\u201315.","journal-title":"J. Nonlin. Sci."},{"key":"7_CR41","doi-asserted-by":"publisher","first-page":"1126","DOI":"10.1086\/147463","volume":"136","author":"JM Mihaljan","year":"1962","unstructured":"Mihaljan, J.M. (1962) A rigorous exposition of the Boussinesq approximation applicable to a thin layer of fluid. Astrophys. J.\n                136, 1126\u20131133.","journal-title":"Astrophys. J."},{"key":"7_CR42","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-08539-4","volume-title":"Mathematical Biology.","author":"JD Murray","year":"1989","unstructured":"Murray, J.D. (1989) Mathematical Biology. Springer, Berlin."},{"key":"7_CR43","doi-asserted-by":"crossref","first-page":"2536","DOI":"10.1103\/PhysRevA.38.2536","volume":"A38","author":"M. Neveling","year":"1988","unstructured":"Neveling, M. & Dangelmayr, G. (1988) Bifurcation analysis of interacting stationary modes in thermohaline convection. Phys. Rev. A38, 2536\u20132543.","journal-title":"Phys. Rev."},{"key":"7_CR44","doi-asserted-by":"crossref","unstructured":"Newell, A.C. (1985) Solitons in Mathematics and Physics, SIAM, CBMS Lecture Series.","DOI":"10.1137\/1.9781611970227"},{"key":"7_CR45","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1017\/S0022112069000176","volume":"38","author":"AC Newell","year":"1969","unstructured":"Newell, A.C. & Whitehead, J.A. (1969) Finite bandwidth, finite amplitude convection. J. Fluid Mech.\n                38, 279.","journal-title":"J. Fluid Mech."},{"key":"7_CR46","volume-title":"Self-Organization in Nonequilibrium Systems","author":"G Nicolis","year":"1977","unstructured":"Nicolis, G. & Prigogine, I. (1977) Self-Organization in Nonequilibrium Systems. Wiley, New York."},{"key":"7_CR47","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4350-2","volume-title":"Applications of Lie Groups to Differential Equations","author":"PJ Olver","year":"1993","unstructured":"Olver, P.J. (1993) Applications of Lie Groups to Differential Equations. Springer-Verlag, New York."},{"key":"7_CR48","unstructured":"Pinho, E.M. (1998) Simetrias Escondidas em Equa\u00e7\u00f5es Diferenciais Parciais. MSc Thesis, University of Porto."},{"key":"7_CR49","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1017\/S0022112069000127","volume":"38","author":"LA Segel","year":"1969","unstructured":"Segel, L.A. (1969) Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mech.\n                38, 203\u2013224.","journal-title":"J. Fluid Mech."},{"key":"7_CR50","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1017\/S0022112089000443","volume":"199","author":"F Simonelli","year":"1989","unstructured":"Simonelli, F. & Gollub, J.P. (1989) Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech.\n                199, 471.","journal-title":"J. Fluid Mech."},{"key":"7_CR51","doi-asserted-by":"publisher","first-page":"1177","DOI":"10.1016\/0094-5765(77)90096-0","volume":"4","author":"GI Sivashinsky","year":"1977","unstructured":"Sivashinsky, G.I. (1977) Nonlinear analysis of hydrodynamic instability in flames-I. Derivation of the basic equations. Acta Astronautica\n                4, 1177\u20131206.","journal-title":"Acta Astronautica"},{"key":"7_CR52","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1098\/rstb.1952.0012","volume":"237","author":"A Turing","year":"1952","unstructured":"Turing, A. (1952) The Chemical Basis of Morphogenesis. Phil. Trans. Roy. Soc. Lond. B 237, 37.","journal-title":"Phil. Trans. Roy. Soc. Lond. B"}],"container-title":["The IMA Volumes in Mathematics and its Applications","Pattern Formation in Continuous and Coupled Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-1-4612-1558-5_7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,30]],"date-time":"2020-01-30T11:44:29Z","timestamp":1580384669000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-1-4612-1558-5_7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999]]},"ISBN":["9781461271925","9781461215585"],"references-count":52,"URL":"https:\/\/doi.org\/10.1007\/978-1-4612-1558-5_7","relation":{},"ISSN":["0940-6573"],"issn-type":[{"type":"print","value":"0940-6573"}],"subject":[],"published":{"date-parts":[[1999]]}}}