linear stability

Thread View: sci.math.num-analysis

2 messages

[T] Back to Threads [V] Tree View [G] Group Articles

2 total messages Started by Ioannis Androulakis Fri, 07 Aug 1992 19:28

#211

Author: Ioannis Androulakis
Date: Fri, 07 Aug 1992 19:28

14 lines
1197 bytes

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 ?
(i.e. let's say that the linearized stability requires
h <= h1, and by running a simulation of the system we
result in h <= h2, with h1 < h2, the starting points of for
the simulation are arbitrarily depending on the accuracy of the
machine..., close to the equilibrium point.)
I know that linear stability analysis precludes such a possibility.

Thank you,
ioannis

Re: linear stability

#224

Author: Andrew Mullhaupt
Date: Mon, 10 Aug 1992 01:35

22 lines
1539 bytes

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

Thread Navigation

This is a paginated view of messages in the thread with full content displayed inline.

Messages are displayed in chronological order, with the original post highlighted in green.

Use pagination controls to navigate through all messages in large threads.

Back to All Threads