  We present a complete dynamical description of the higher-order modulation
instability for a fourth-order nonlinear Schr\"{o}dinger equation. For
two-breather solutions of this equation, we have identified the locus in a
geometrical space where the growth rates for the breathers are equal in
parameter space. We show that a circle bounds the entire parameter space for
the nonlinear Schr\"{o}dinger equation. In contrast, it is bound by an
intersecting circle and an ellipse for the fourth-order equation. We show that,
for all the higher-order equations in the nonlinear Schr\"{o}dinger equation
hierarchy, the parameter space follows a similar geometric interpretation as
for the fourth-order equation.
