Analyzing Hyperbolic Dynamics of Maps x_{n+1}=Ax_n

[standardflop]

Hello,
Given the three maps $x_{n+1}=Ax_n$ with
$$A_1=\begin{pmatrix} 1&-1\\1&1 \end{pmatrix}, A_2=\begin{pmatrix} 1/2&1/2\\-1&1 \end{pmatrix}, A_3=\begin{pmatrix} 3&2\\5/2&2 \end{pmatrix},$$
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.

 

[Phlogistonian]

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

 

[Moetasim]

I would like to provide a detailed response to the content regarding the hyperbolic dynamics of maps. First, let's define what hyperbolic dynamics means. In mathematics, a dynamical system is considered hyperbolic if its behavior is chaotic and sensitive to initial conditions. In other words, small changes in the initial conditions can lead to drastically different outcomes. This is often characterized by the presence of unstable periodic orbits and chaotic behavior.

Now, let's analyze the three maps given in the content. A1 is a 2x2 matrix with eigenvalues of 1 and 2. As mentioned, A1 is a hyperbolic map because none of its eigenvalues lie on the unit circle. This means that the behavior of this map is chaotic and sensitive to initial conditions.

A2 is also a 2x2 matrix but with eigenvalues of 1/2 and 1. Since one of its eigenvalues (1) lies on the unit circle, A2 is not a hyperbolic map. Its behavior may still exhibit some chaotic behavior, but it is not as sensitive to initial conditions as A1.

Now, let's focus on A3, which has eigenvalues of 3 and 2. The question posed in the content is whether a system with eigenvalues {L1,L2} such that 0<L1<1<L2 is hyperbolic or not. The answer is not straightforward. In general, the eigenvalues of a hyperbolic map should be either both greater than 1 or both less than 1. However, there are cases where this rule may not apply. In the case of A3, the eigenvalues are 3 and 2, where 0<2<3. This means that A3 may exhibit some hyperbolic behavior, but it cannot be classified as a hyperbolic map based on its eigenvalues alone.

The presence of complex eigenvectors in A1 and A2 does not necessarily say anything about the general dynamics. However, in the case of A3, the complex eigenvectors can provide information about the stable and unstable subspaces. The eigenvectors of A3 tell us the direction of the stable and unstable manifolds, which can give us a better understanding of the dynamics of this map.

In conclusion, the dynamics of a map can be hyperbolic if its eigenvalues do not lie on the unit circle. However, there