Question

Solve each item supporting your answer with clear expalanation.

1. Consider the dynamical system

x₁(t + 1) = 0.1x₁(t) + 0.2x₂(t) + 1

x₂(t + 1) = 0.4x₁(t) + 0.3x₂(t) + 2

a. Find the closed formula for the vector x(t).

b. Find the equilibrium state of this system and determine its stability.

Answer 1

Given, x₁(t + 1) = 0.1 x₁ (t) + 0.2 x₂(t) + 1

x₂(t+1) = 0.4 x₂ (t) + 0.3 x₂(t) + 2

a) For homogeneous solution

eigen value of  

eigenvector corresponding be V

R₂ R₂ + R₁

_Let  V_{2 =} 1 , V₁ = 0.5

V =

Eigenvector corresponding  

R₂   R₂ - 2R₁

Let m₂ = 1 , m₁ = -1

m =

X^H (t) = C₁ (0.5)^t   

0.9A - 0.2B = 1

-0.4A + 0.7B = 2

A = 2 , B = 4

X(t) = C₁ (0.5)^t  

b) For euilibrium solution X₁ (t + 1) = X₁(t) , X₂ (t + 1) = X₂(t)

OR WE SAY   

X(t) =

Since both eigen value || < 1

So it is a stable answer.