Nilpotent Jacobians and Almost Global Stability

Abstract

By one hand, we continue with the study of the liaison between the almost Hurwitz vector fields and density functions. In particular by using mapsHwith nilpotent JacobianJHsuch that their rows are linearly dependent over R, we construct a family of almost Hurwitz vector fieldsFin dimension larger than two which has the origin as almost global attractor. This last fact is shown by associating an appropriate density function to the vector fieldF. Moreover, we show new examples of Hurwitz vector fields such that the origin is a global attractor. On the other hand, in the case when the rows ofJHare linearly independent over R, we show explicitly the inverse maps of the counterexamples to Generalized Dependence Problem and proving that this inverse maps preserve the linearly independence over R of the nilpotent Jacobian.