Rebuttal of “Existence of attractor and control of a 3D differential system” by Z. Wang

Abstract

A paper, “Existence of attractor and control of a 3D differential system,” was published in the journal Nonlinear Dynamics. The author tries to prove the existence of horseshoe chaos by means of the Shilnikov criterion. To do that, he uses the undetermined coefficient method to analytically demonstrate the existence of heteroclinic orbits in a Lorenz-like system. Unfortunately, his proof is not correct for two reasons. Firstly, he considers an odd function to represent the heteroclinic orbit whereas such an orbit joining two saddle-focus equilibria can never have that symmetry. Secondly, he looks for a structurally unstable Shilnikov heteroclinic orbit by means of uniformly convergent series expansions: This would imply that the dynamical object found is structurally stable.