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Chapter 7: Q25E (page 324)

Consider a dynamical system x鈫�(t+1)=Ax鈫�(t)with two components. The accompanying sketch shows the initial state vector x鈫�0and two eigen vectors 蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫�of A (with eigen values 位1鈫�and位2鈫�respectively). For the given values of 位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=1,位2鈫�=0.9

Short Answer

Expert verified

So, the required solution is Atx0=伪蠀1+0.9t尾蠀2.

Step by step solution

01

Define the eigenvector 

Eigenvector:An eigenvector of Ais a nonzero vector vin role="math" localid="1668402410209" Rnsuch that role="math" localid="1668402421328" Av=位v, for some scalar 位.

02

Note the given data

It is given that:

位1鈫�=1,位2鈫�=0.9

Given graph is:

03

Finding the required solution

We have:

A蠀1=蠀1A蠀2=0.9蠀2

For x0=伪蠀1+尾蠀2,We have:

Ax0=A(伪蠀1+尾蠀2)=伪A蠀1+尾A蠀2=伪蠀1+0.9尾蠀2

Therefore, Atx0=伪蠀1+0.9t尾蠀2.

Hence, the solutions is Atx0=伪蠀1+0.9t尾蠀2.

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