Patterns on growing square domains via mode interactions

References

• Turing , AM . 1952 . The chemical basis of morphogenesis . Phil. Trans. Roy. Soc. Lond. B , 237 : 37 – 72 .
   Web of Science ®Google Scholar
• Castets , V . 1990 . Experimental evidence of a sustained Turing-type equilibrium chemical pattern . Phys. Rev. Lett. , 64 ( 3 ) : 2953 – 2956 .
   PubMed Web of Science ®Google Scholar
• Ouyang , Q and Swinney , HL . 1991 . Transition from a uniform state to hexagonal and striped Turing patterns . Nature , 352 : 610 – 612 .
   Web of Science ®Google Scholar
• Srinivasan , S , Rashka , KE and Bier , E . 2002 . Creation of a Sog morphogen gradient in the Drosophila embryo . Dev. Cell , 2 : 91 – 101 .
   PubMed Web of Science ®Google Scholar
• Baringa , M . 1994 . Looking to development's future . Science , 266 : 561 – 564 .
   Web of Science ®Google Scholar
• Crampin , EJ , Gaffney , EA and Maini , PK . 1999 . Reaction and diffusion on growing domains: Scenarios for robust pattern formation . Bull. Math. Biol. , 61 ( 6 ) : 1093 – 1120 .
   PubMed Web of Science ®Google Scholar
• Crampin , EJ . Reaction-diffusion Patterns on Growing Domains, PhD Thesis , Magdalen College, University of Oxford, 2000
   Google Scholar
• Plaza , R . 2004 . The effect of growth and curvature on pattern formation . J. Dynam. Diff. Eqns. , 16 ( 4 ) : 1093 – 1121 .
   Google Scholar
• Kondo , S and Asai , R . 1995 . A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus . Nature , 376 : 765 – 768 .
   PubMed Web of Science ®Google Scholar
• Crampin , EJ , Gaffney , EA and Maini , PK . 2002 . Mode doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model . J. Math. Biol. , 44 ( 2 ) : 107 – 128 .
   PubMed Web of Science ®Google Scholar
• Madzvamuse , A , Wathen , AJ and Maini , PK . 2003 . A moving grid finite element method applied to a model biological pattern generator . J. Comp. Phys. , 190 : 478 – 500 .
   Web of Science ®Google Scholar
• 2005 . A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains . J. Sci. Comput. , 24 ( 2 ) : 247 – 262 .
   Web of Science ®Google Scholar
• Izhikevich , E . 2000 . Neural excitability, spiking and bursting . Int. J. Bif. Chaos , 10 ( 6 ) : 1171 – 1266 .
   Web of Science ®Google Scholar
• Golubitsky , M , Josic , K and Kaper , TJ . 2001 . “ An unfolding theory approach to bursting in fast-slow systems ” . In Global Analysis of Dynamical Systems: Festschrift dedicated to Floris Takens on the occasion of his 60th birthday , Edited by: Broer , HW , Krauskopf , B and Vegter , G . 277 – 308 . Bristol : Institute of Physics Publ .
   Google Scholar
• Crawford , JD . 1991 . “ Boundary conditions as symmetry constraints ” . In Singularity Theory and Its Applications, Warwick 1989, Part II. , Edited by: Roberts , M and Stewart , IN . Vol. 1463 , 63 – 79 . Heidelberg : Springer-Verlag Lecture Notes in Math .
   Google Scholar
• Crawford , JD . 1991 . Surface waves in non–square containers with square symmetry . Phys. Rev. Lett. , 67 : 441 – 444 .
   PubMed Web of Science ®Google Scholar
• Golubitsky , M and Stewart , I . 2002 . The Symmetry Perspective , Vol. 200 , Basel : Progress in Mathematics, Birkhäuser .
   Google Scholar
• Crawford , JD , Gollub , JP and Lane , D . 1993 . Hidden symmetries of parametrically forced waves . Nonlinearity , 6 : 119 – 164 .
   Web of Science ®Google Scholar
• Golubitsky , M , Stewart , IN and Schaeffer , DG . 1988 . Singularities and Groups in Bifurcation Theory: , Vol. 69 , New York : Applied Mathematical Sciences, Springer-Verlag .
   Google Scholar
• Crawford , JD . 1996 . “ D4-symmetric Maps with hidden Euclidean symmetry ” . In Pattern Formation and Symmetry Breaking in PDE's , Edited by: Chadan , J , Golubitsky , M , Langford , W and Wetton , B . Vol. 5 , Providence : Fields Institute Communications, AMS .
   Google Scholar
• Swift , JW . Bifurcation and Symmetry in Convection, PhD Thesis, , University of California, Berkeley 1984
   Google Scholar
• Proctor , MRE and Matthews , PC . 1996 . Resonance in non–Boussinesq convection . Physica D , 97 : 229 – 241 .
   Web of Science ®Google Scholar
• Golubitsky , M , Stewart , IN and Schaeffer , DG . 1984 . Singularities and Groups in Bifurcation Theory , Vol. 51 , New York : Applied Mathematical Sciences, Springer-Verlag .
   Google Scholar