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Classify the origin as an attractor, repeller, or saddle point of the dynamical system $\mathbf{x}_{k+1}=A \mathbf{x}_{k} .$ Find the directions of greatest attraction and/or repulsion.$A=\left[\begin{array}{rr}{.4} & {.5} \\ {-.4} & {1.3}\end{array}\right]$

There is no eigenvalue that is greater than 1 in magnitude, so no direction of greatest repulsion.

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 6

Discrete Dynamical Systems

Vectors

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Okay. We want to solve this. Only e using this initial condition with a physical to this. So the first thing that we do is to find a iving values so determined. Oh, a month Land our identity. So one minus dander. Minus four. Mice land, uh, minus two and three. So then this is equal to minus one. Lambda is foremost land law plus six to this equal to them squared. Plus Amanda minus a plus. Plus. Pull, Amber. Wanna stand though? Minus four plus six. This is Lando. Swear plus three, huh? 03 llama plus two. Take it. Land a minus plus one. And the prostitute? The service is equal to zero. You have Lando exactitude minus one. Old mines, too. So what this means is that we have on a tractor now to find the island values. I mean, Victor's, we need to Seoul a minus. Land our identity. That's why Geeta Physics. So let's start with you on this case. So now let's do lamb. The easier to minus one. We have Well, a is equal to 13 minus two and minus four. So we have one minus minus one. So let's give you two months to throwing into four minus minus one. So 84 +123 kinds of pita. You know, 12 uses, right? Taking the first right of the first caller. Yes to you. No one wants to eat a too busy to throw. So you don't want is even to eat it too. Now, let's eat out. One is one. So you have either is equal to one and one such a first island mountain. Now, for your second ardent victor, I'm not gonna do it here, and you can do it yourself. But you are in Victor's. Our land, not one is equal to minus one they want is equal to 11 land. That, too, is equal to negative to be too. You think you two and three? Yeah. Now your general solution to your early E is then X is a function of tea. Is it good to see someone? One on one eats the most one or medium iced tea. Plus. See some too. 23 eats the minus two tea. See? So now we use our initial condition. Zero is equal to 32 So 32 he's equal to you. Said 111 plus season to season two Cheers through. So then you have three Easy to cease of one plus two cities up to and to use it to see someone plus three c sub. Sure. Yeah. Now, subtracting these equations you have one is able to true seize up to minus three seats. Up to is negative, Caesar two. So seize up to his ex connected one. And that would imply that Season one, this one is equal to three minus, minus two. Look, now you see someone, you see three executives see someone minus two to see someone is X file. So get general solution. It is. Then it's a function too. These people to you. Five climbs 11 each of the minus T loss minus one less minus. One times 23 each, The minus Sure to

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