🚀 go-pugleaf

In article <androula.713215737@lips.ecn.purdue.edu> androula@lips.ecn.purdue.edu (Ioannis Androulakis) writes:
>Given a discrete nonlinear dynamical system that depends
>on a parameter (say h), is it possible that the linearized stability
>around that equilibrium point results in a value of the
>parameter that differs from the value that is identified
>by simulating the dynamics of the system on a computer ?

Yes. Suppose that in the nonlinear system, the unstable manifold of the fixed
point is "re-injected" by the nonlinear dynamics into the stable manifold of
the fixed point. Then the linear stability analysis will not be able to explain
the apparent stability of the nonlinear simulations. Now as you continue to
push your parameter, this re-injection may involve a large enough excursion
for the simulation to notice.

>I know that linear stability analysis precludes such a possibility.

To justify such analysis you can appeal to Hartman-Grobman near hyperbolic
fixed points, and beyond that it's pretty interesting. If your fixed point
isn't hyperbolic, you need center manifold theory.

Later,
Andrew Mullhaupt