Turing patterns in three dimensions.

We investigate three-dimensional Turing patterns in two-component reaction diffusion systems. The FitzHugh-Nagumo equation, the Brusselator, and the Gray-Scott model are solved numerically in three dimensions. Several interconnected structures of domains as well as lamellar, hexagonal, and spherical domains are obtained as stable motionless equilibrium patterns. The relative stability of these structures is studied analytically based on the reduction approximation. The relation with the microphase-separated structures in block copolymers is also discussed.

[1]  J. E. Pearson.  Complex Patterns in a Simple System , 1993, Science.

[2]  De Wit A,et al.  Twist grain boundaries in three-dimensional lamellar Turing structures. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[3]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[4]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[5]  Michael L. Kagan,et al.  New systems for pattern formation studies , 1992 .

[6]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability , 1983 .

[7]  G. Dewel,et al.  Fluctuations near nonequilibrium phase transitions to nonuniform states , 1980 .

[8]  From quasi-2D to 3D Turing patterns in ramped systems , 1996 .

[9]  I. Prigogine,et al.  Symmetry Breaking Instabilities in Dissipative Systems. II , 1968 .

[10]  I. Prigogine,et al.  Symmetry Breaking Instabilities in Biological Systems , 1969, Nature.

[11]  Milos Dolnik,et al.  Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes. , 2002, Physical review letters.

[12]  J. Boissonade,et al.  Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system , 1980 .

[13]  Decay of Metastable Rest State in Excitable Reaction-Diffusion System , 1989 .

[14]  Periodic spatiotemporal patterns in a two-dimensional two-variable reaction-diffusion model. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  E. Dulos,et al.  Stationary Turing patterns versus time-dependent structures in the chlorite-iodide-malonic acid reaction , 1992 .

[16]  H. Nakano,et al.  Nematic-Smectic Transition in an Aligned Rod System , 1979 .

[17]  Dulos,et al.  Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. , 1990, Physical review letters.

[18]  M. Cross,et al.  Turing instability in a boundary-fed system , 1998, patt-sol/9806006.

[19]  Harry L. Swinney,et al.  Transition to chemical turbulence. , 1991, Chaos.

[20]  Raymond Kapral,et al.  Microscopic model for FitzHugh-Nagumo dynamics , 1997 .

[21]  D. Morse,et al.  Orthorhombic Fddd network in triblock and diblock copolymer melts. , 2005, Physical review letters.

[22]  David Andelman,et al.  Phase transitions in Langmuir monolayers of polar molecules , 1987 .

[23]  Kimmo Kaski,et al.  A new dimension to Turing patterns , 2002, cond-mat/0211283.

[24]  Aleksij Aksimentiev,et al.  Morphology of surfaces in mesoscopic polymers, surfactants, electrons, or reaction-diffusion systems: Methods, simulations, and measurements , 2002 .

[25]  H. Swinney,et al.  Transition from a uniform state to hexagonal and striped Turing patterns , 1991, Nature.

[26]  Y. Oono,et al.  Spinodal decomposition in 3-space. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  Sebastian Doniach,et al.  Phase transitions with spontaneous modulation-the dipolar Ising ferromagnet , 1982 .

[28]  J. Keener,et al.  Singular perturbation theory of traveling waves in excitable media (a review) , 1988 .

[29]  Michael C. Cross,et al.  Numerical bifurcation diagram for the two-dimensional boundary-fed chlorine-dioxide–iodine–malonic-acid system , 1998, patt-sol/9806007.

[30]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[31]  Stéphane Métens,et al.  Pattern selection in bistable systems , 1997 .

[32]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[33]  K. Kawasaki,et al.  Equilibrium morphology of block copolymer melts , 1986 .

[34]  Edgar Knobloch,et al.  Pattern formation in the three-dimensional reaction-diffusion systems , 1999 .

[35]  J. Boissonade,et al.  Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction , 1991 .

[36]  Teemu Leppänen,et al.  Morphological transitions and bistability in Turing systems. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Takao Ohta,et al.  Fddd structure in AB-type diblock copolymers , 2006, Journal of physics. Condensed matter : an Institute of Physics journal.

[38]  Takao Ohta,et al.  Interconnected Turing patterns in three dimensions. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  S. Métens,et al.  Pattern formation and spatial self-entrainment in bistable chemical systems. , 2002, Faraday discussions.

[40]  J Rinzel,et al.  Traveling wave solutions of a nerve conduction equation. , 1973, Biophysical journal.

[41]  Ryo Kobayashi,et al.  Higher-dimensional localized patterns in excitable media , 1989 .

[42]  Zoltán Noszticzius,et al.  Spatial bistability of two-dimensional turing patterns in a reaction-diffusion system , 1992 .

[43]  Takao Ohta,et al.  Kinetics of morphological transitions in microphase-separated diblock copolymers , 2004 .

[44]  Takashi Teramoto,et al.  Double Gyroid Morphology in a Gradient System with Nonlocal Effects , 2002 .

[45]  Guy Dewel,et al.  Three-dimensional dissipative structures in reaction-diffusion systems , 1992 .

[46]  Teemu Leppänen,et al.  Turing systems as models of complex pattern formation , 2004 .