Global dynamics of stationary solutions of the extended Fisher-Kolmogorov equation

In this paper we study the fourth order differential equation d4u/dt4 + q d2u/dt2 + u3 - u = 0,, which arises from the study of stationary solutions of the Extended Fisher-Kolmogorov equation. Denoting x = u, y = du/dt, z = d2u/dt2, v = d3u/dt3 this equation becomes equivalent to the polynomial system x = y, ẏ = z, ż = v, v̇ = x - qz - x3 with (x, y, z, v) ε R4 and q ε R and As usual, the dot denotes the derivative with respect to the time t. Since the system has a first integral we can reduce our analysis to a family of systems on R3 We provide the global phase portrait of these systems in the Poincaré ball (i.e., in the compactification of R3 with the sphere S2 of the infinity). © 2011 American Institute of Physics.