  We discuss a relationship between information geometry and the
Glansdorff-Prigogine criterion for stability. For the linear master equation,
we found a relation between the line element and the excess entropy production
rate. This relation leads to a new perspective of stability in a nonequilibrium
steady-state. We also generalize the Glansdorff-Prigogine criterion for
stability based on information geometry. Our information-geometric criterion
for stability works well for the nonlinear master equation, where the
Glansdorff-Prigogine criterion for stability does not work well. We derive a
trade-off relation among the fluctuation of the observable, the mean change of
the observable, and the intrinsic speed. We also derive a novel thermodynamic
trade-off relation between the excess entropy production rate and the intrinsic
speed. These trade-off relations provide a physical interpretation of our
information-geometric criterion for stability. We illustrate our
information-geometric criterion for stability by an autocatalytic reaction
model, where dynamics are driven by a nonlinear master equation.
