The work is based on a general theory that helps to derive normal forms of dynamical systems from a system of spatially isotropic three-dimensional coupled nonlinear partial differential equations of the reaction-diffusion type in the case when its zero solution loses stability against finite-wavelength perturbations (the Turing instability). Assuming periodic boundary conditions, the dynamical systems are generated by the truncation of the PDEs, projecting them onto a center manifold corresponding to a set of spatial harmonics based on three, four, or six different vectors. The projection is performed, using the corresponding representations of the isotropy group in the three-dimensional space. The three above-mentioned cases are referred to, respectively, as the simple cubic, face-centered cubic, and body-centered cubic (BCC) lattices. In the latter case, the derived normal form contains not only cubic, but also quadratic nonlinear terms. It is found that, in some cases, the consistent analysis demands higher-order (quintic) nonlinear terms to be added (which is not actually done in this work). In all the three cases, the normal forms of the dynamical systems depend on a single control parameter.
For two particular reaction-diffusion models, viz., the Brusselator and Lengyel-Epstein ones, the control parameter is calculated as a function of actual parameters of the models. A chain of bifurcation diagrams is analyzed for each dynamical system, varying the control parameter. Different stationary solutions appearing in the diagrams are interpreted as solutions, with a certain spatial symmetry, to the underlying PDE system. In the dynamical system corresponding to the BCC lattice, all the stationary solutions are found to be unstable near the onset of the Turing instability; however, in a case when the coefficient in front of the quadratic nonlinear terms is small, that is relevant to the applications, secondary bifurcations occur close to the onset, stabilizing various solutions. In fact, this mechanism is a counterpart of the secondary-bifurcation scenario well-known in two-dimensional pattern-forming models combining cubic and quadratic nonlinearities. It is found that up to four different stationary solutions stabilized by the secondary bifurcations may coexists in the BCC system. The obtained results are compared to earlier published direct simulations of the Brusselator model in the three-dimensional geometry.