On the global dynamics of the Newell–Whitehead system. Issue 4 (2nd October 2019) - Valls, Claudia
Abstract : In this paper by using the Poincaré compactification in ℝ 3 we make a global analysis of the model x ′ = z, y ′ = b ( x – dy ), z ′ = x ( x 2 – 1)+ y + cz with b ϵ ℝ and c, d ϵ ℝ +, here known as the three-dimensional Newell–Whitehead system. We give the complete description of its dynamics on the sphere at infinity. For some values of the parameters this system has invariant algebraic surfaces and for these values we provide the dynamics of the system restricted to these surfaces and its global phase portrait in the Poincaré ball. We also include the description of the α -limit and ω -limit set of its orbits in the Poincaré ball including its boundary, that is, in the compactification of ℝ 3 with the sphere at the infinity. We recall that the restricted systems are not analytic and so in this paper we overcome this difficulty by using the blow-up technique.