Spatial structures in a reaction-diffusion system--detailed analysis of the "Brusselator".

Continuous dependence of spatially nonuniform concentration profiles for the 'Brussellator" reaction mechanims on the characteristic length of the system is given both for zero flux and fixed boundary conditions. Branches of solutions arising through primary bifurcation form closed curves. Secondary bifurcations giving rise to spatially asymmetric solutions exist for fixed boundary conditions. Results of a stability analysis of individual solutions are discussed. A method of composing complex spatial profiles for higher lengths from elementary solutions for smaller lenghts is suggested and tested in the case of zero flux boundary conditions. Emergence of subsequently more complex stable patterns in dependence on increasing length of the system suggests many similarities to gradual build up of complex morphogenetic patterns.

[1]  J. Keener.  SECONDARY BIFURCATION IN NONLINEAR DIFFUSION REACTION EQUATIONS. , 1976 .

[2]  I. Prigogine,et al.  On symmetry-breaking instabilities in dissipative systems , 1967 .

[3]  R. Lefever.  Dissipative Structures in Chemical Systems , 1968 .

[4]  G. Nicolis,et al.  Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equations and the steady state solutions , 1975 .

[5]  K. Kirchgässner.  Bifurcation in Nonlinear Hydrodynamic Stability , 1975 .

[6]  Steady state spatial structures in dissipative systems: Numerical algorithm and detailed analysis , 1977 .

[7]  D. Cohen,et al.  Bifurcation of Localized Disturbances in a Model Biochemical Reaction , 1976 .

[8]  I. Stakgold,et al.  Branching of Solutions of Nonlinear Equations , 1971 .

[9]  Edward L. Reiss,et al.  Multiple Eigenvalues Lead to Secondary Bifurcation , 1975 .

[10]  J. Tyson.  Some further studies of nonlinear oscillations in chemical systems , 1973 .

[11]  M. A. Collins,et al.  Inhomogeneous stationary states in reaction-diffusion systems. , 1976, Biophysical chemistry.

[12]  M. Marek,et al.  Nonlinear phenomena in oscillatory systems of homogeneous reactions - experimental observations. , 1975, Biophysical chemistry.

[13]  M. Herschkowitz-Kaufman.  Bifurcation analysis of nonlinear reaction-diffusion equations—II. Steady state solutions and comparison with numerical simulations , 1975 .

[14]  Grégoire Nicolis,et al.  Localized Spatial Structures and Nonlinear Chemical Waves in Dissipative Systems , 1972 .

[15]  Louis Bauer,et al.  NUMERICAL BIFURCATION AND SECONDARY BIFURCATION: A CASE HISTORY**This research was supported by the National Science Foundation under Grant GP-27223 and the Office of Naval Research under Contract N00014-67-A-0467-0006. The computations were performed at the ERDA Computing and Applied Mathematics Ce , 1976 .