Variety of Evolutions to Stationary Periodical Structures

Abstract

The system (1) with S-shaped f(u,v) =0 and bifurcations from one stationary state to trigger, and from trigger to another stationary state (in the subsystem u,v) induced by φ(p)

$$\begin{array}{l} {u_{t}} - {D_{u}}{u_{{XX}}} = {\varepsilon _{1}}f\left( {u,v} \right) \\ {v_{t}} - {D_{v}}{v_{{XX}}} = {\varepsilon _{2}}\left( {g\left( {u,v} \right) + \varphi \left( p \right)} \right) \\ {p_{t}} - {D_{p}}{p_{{XX}}} = {\varepsilon _{3}}\left( {\alpha u - \beta p} \right)\quad {\varepsilon _{1}} \gg {\varepsilon _{3}} \gg {\varepsilon _{2}} \\ \end{array}$$

((1))

gives different types of evolutions to stationary periodical structures for proper initial conditions and values of parameters.