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

Abstract

In this paper we study the fourth order differential equation d(4)u/dt(4) + q d(2)u/dt(2) + u(3) - u = 0, which arises from the study of stationary solutions of the Extended Fisher-Kolmogorov equation. Denoting x = u, y = du/dt, z = d(2)u/dt(2), v = d(3)u/dt(3) this equation becomes equivalent to the polynomial system. (x) over dot = y, (y) over dot = z, (z) over dot = v, (v) over dot = x - qz - x(3) with (x, y, z, v) is an element of R(4) and q is an element of R. 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 R(3). We provide the global phase portrait of these systems in the Poincare ball (i.e., in the compactification of R(3) with the sphere S(2) of the infinity). (C) 2011 American Institute of Physics. [doi: 10.1063/1.3657425]