A New Mechanical Algorithm for Calculating the Amplitude Equation of the Reaction-Diffusion Systems

Turing (1952) showed that under certain conditions, reaction and diffusion processes alone could lead to the symmetry-breaking instability (i.e., Turing instability) of a system (i.e., Turing system) from a homogeneous state to a stationary patterned state (Setayeshgar & Cross, 1999). Turing system can generate stationary patterns such as stripes and/or spots. Turing instability mechanism has particular relevance to pattern formation in nonlinear complex systems. Nowadays, pattern formation is one of the central problems of the natural, social, and technological science (Medvinsky et al., 2002). Especially in population dynamics, pattern can clarify the distribution and development of different species. There are a number of theoretical and numerical studies of pattern formation by using weakly nonlinear analysis (Chen & Vinals, 1999; Dufiet & Boissonade, 1996; Ipsen et al., 1998; Ipsen et al., 2000; Malomed, ABSTRACT