  This paper focuses on the equilibria and their regions of attraction for
continuous-time nonlinear dynamical systems. The classical Poincar\'e--Hopf
Theorem from differential topology is used to derive a result on a sufficient
condition on the Jacobian for the existence of a unique equilibrium for the
system, which is in fact locally exponentially stable. We show how to apply
this result to the deterministic SIS networked model, and a nonlinear
Lotka--Volterra system. We apply the result further to establish an
impossibility conclusion for a class of distributed feedback controllers whose
goal is to drive the SIS network to the zero equilibrium. Specifically, we show
that if the uncontrolled system has a unique nonzero equilibrium (a diseased
steady-state), then the controlled system also has a unique nonzero
equilibrium. Applying results from monotone dynamical systems theory, we
further show that both the uncontrolled and controlled system will converge to
their nonzero equilibrium from all nonzero initial conditions exponentially
fast. A counterpart sufficient condition for the existence of a unique
equilibrium for a nonlinear discrete-time dynamical system is also presented.
