Transition from polar to duplicate patterns.

The existence of symmetric nonuniform solutions in nonlinear reaction-diffusion systems is examined. In the first part of the paper, we establish systematically the bifurcation diagram of small amplitude solutions in the vicinity of the two first bifurcation points. It is shown that: i)The system can adopt a stable symmetric solution (basic wave number 2) if the value of the bifurcation parameter is changed or if the initial polar structure (basic wave number 1) is sufficiently perturbed. ii) This behavior is independent of the particular reaction-diffusion model proposed and of the number of intermediate components (greater than or equal to 2) involved. In the second part of the paper, analogies are established between the possibilities offered by the bifurcation diagrams, involving only the two first primary branches, the observation that in the early development of different organisms, appropraite experimental manipulations may switch the normal (polar) developmental pattern to a duplicate structure.