• Support PF! Buy your school textbooks, materials and every day products Here!

Hyperbolic dynamics.

Hello,
Given the three maps [itex] x_{n+1}=Ax_n[/itex] with
[tex] A_1=\begin{pmatrix} 1&-1\\1&1 \end{pmatrix}, A_2=\begin{pmatrix} 1/2&1/2\\-1&1 \end{pmatrix}, A_3=\begin{pmatrix} 3&2\\5/2&2 \end{pmatrix}, [/tex]
describe the dynamics, and say whether or not the dynamics is hyperbolic.

Finding eigenvalues and eigenvectors is relatively easy. And i know that the dynamics is hyperbolic if no eigenvalue lies on the unit circle (A1 is thus hyperbolic, and A2 is not). But is a system with eigenvalues {L1,L2} such that 0<L1<1<L2 hyperbolic (A3), or should both eigenvalues numerically be either greater or less than 1 ?

Also, does complex eigenvectors say anything about the general dynamics (cases: A1,A2) or is all that you can conclude that the origin is an unstable spiral because Re(Li)>0 ? I mean, in the case A3, the eigenvectors tells us the eigendirection of the stable and unstable subspaces, right?

Thanks in advance.
 

Answers and Replies

I think we need more information. I don't know what is meant by "dynamics" in this problem.
 

Related Threads for: Hyperbolic dynamics.

  • Last Post
Replies
2
Views
1K
  • Last Post
2
Replies
44
Views
6K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
903
  • Last Post
Replies
1
Views
10K
  • Last Post
2
Replies
26
Views
791
  • Last Post
Replies
1
Views
666
  • Last Post
Replies
1
Views
694
Top