  We examine the phenomenon of nonlinear stabilization, exhibiting a variety of
related examples and counterexamples. For G\^ateaux differentiable maps, we
discuss a mechanism of nonlinear stabilization, in finite and infinite
dimensions, which applies in particular to hyperbolic partial differential
equations, and, for Fr\'echet differentiable maps with linearized operators
that are normal, we give a sharp criterion for nonlinear exponential
instability at the linear rate. These results highlight the fundamental open
question whether Fr\'echet differentiability is sufficient for linear
exponential instability to imply nonlinear exponential instability, at possibly
slower rate.
