Re: Question about global stablity?

From: Fan Yang (yang_at_cae.wisc.edu)
Date: 11/19/04 

Date: Thu, 18 Nov 2004 22:44:03 -0600

Dear James,

Thanks for your reply.

However, your example doen not satisfy my assumption. If all the
eigenvalues are less than -1, then there should be no periodic
orbit (r=1). Am I correct?

Thanks,

Fan

"James Meiss" <jdm@NOSPAM.invalid> wrote in message
news:jdm-9DE646.17440918112004@peabody.colorado.edu...
> In article <cnjc33$3m8$1@news.doit.wisc.edu>,
> "Fan Yang" <yang@cae.wisc.edu> wrote:
>
>> Dear Thoms,
>>
>> Thanks a lot for your reply. :-)
>>
>> First, I do mean all the eigenvalues are real and < -1. It implies
>> the local asymptotical stabality because we are in continuous
>> time ODE formulation.
>>
>> Right, you gave a good example. However, the eigenvalue
>> of x=1/2 is 1/2. It doesn't satisfy our assumption that the Jacobian
>> matrix has all the eigenvalues less than -1 for all x. Btw, we can
>> also make the assumption that there exists only ONE fixed point.
>> Under this condition, can we infer the global A.S. of this fixed
>> point?
>>
>
> No. For example
>
> d/dt r= -r(1-r)
> d/dt theta = f(r)
>
> Has an asymptotically stable fixed point at the origin (r=0), and no
> other fixed points (if f(r) is nonzer0 everywhere), but an unstable
> periodic orbit at r=1, and infinity is an attractor.
>
> --
> Jim Meiss
> <http://amath.colorado.edu/faculty/jdm>