Necessary and sufficient conditions of instability of nonlinear autonomous dynamic systems (Q1178564)

The paper deals with the study when an equilibrium point \(x_ 0\) for an autonomous differential equation \(\dot x=f(x)\), i.e. \(f(x_ 0)=0\), is unstable. Specifically, it is shown that when there exists a scalar function \(\mu(x)\) which is positive in some neighborhood and for which the divergence \(\hbox{div}[\mu(x)f(x)]>0\) on an open part of this neighborhood, then necessarily the equilibrium \(x_ 0\) is unstable. The idea of using such an ``absorbtion'' function \(\mu\) is extended to parametrized functions \(\mu(x,\nu)\) with \(\nu\) being a scalar parameter; and the corresponding positive divergence condition is adapted in an appropriate way.