Loss of Attractivity in the Generalization from 1D to 2D: Case Studies

In this note, we investigate an open stability question in the field of multidimensional systems. In the linear case, we know that attractivity implies stability, but this result is not true for nonlinear systems with famous counterexamples. We show here that generalizing these counterexamples from the 1D to the 2D case is not straightforward, as the stability properties are not preserved. Two models are studied in this article, one in the discrete case and the other in the continuous case, but it remains an open question whether attractivity can imply stability in a nonlinear 2D framework.