Toronto Math Forum

MAT244-2014F => MAT244 Math--Lectures => Topic started by: Li on December 05, 2014, 05:22:42 PM

Title: How can I decide what type of the local phase portrait is
Post by: Li on December 05, 2014, 05:22:42 PM
firstly, if the linear system I solved is center, how can I decide the locally linear system is center or spiral?
secondly, if the linear system I solved is two equal real eigenvalue, how can I decide wether it is node or spiral points?
I did the problem on the text book, in the answer it just said it is center or spiral, undetermined, do we need to write the exact type in final?   
Title: Re: How can I decide what type of the local phase portrait is
Post by: Victor Ivrii on December 05, 2014, 06:26:19 PM
Generally from linearized system you cannot tell center from a spiral point. So your correct answer should be "it is either center or a spiral point" but you also should indicate  if it is clock-wise or counter-clock-wise oriented.

As I explained if right-hand expressions are smooth and $r_1=r_2\ne 0$ then it will be either a proper node or improper node—of the same type as for a linearized system. But it will not be considered as an error to answer "either node or spiral point" with indication if it is stable or unstable