The Poincar\'e-Bendixson Theorem and the Non-linear Cauchy-Riemann   Equations
  Fiedler and Mallet-Paret prove a version of the classical Poincar\'e-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function. 
J.B. van den Berg and S. Munao and R.C.A.M. Vandervorst
math.AP math.DG math.DS