Explain the determine purple and inf is here and Why did we choose this equilibrium point? Proposition [22] Assume that p,q € R. Then\p| + |q] < 1is a sufficient condition for the asymptotic stability of the difference equationХn+1 — рап — qxn-1 — 0, п 3 0, 1,... .Lemma 1. The equilibrium points of the difference equation (2) are 0 and ±/1- aProof.(x)+ a(7)° + ax = X(7) + (a – 1) = 07{(T) + (a – 1)} = 0This means that the equilibrium points are 0 and ±/1 – a. Motivated by the above studies, we study the dynamics of following higherorder difference equationXn+1A + B(2)п-тwhere A, B are positive real numbers and the initial conditions are positivenumbers. Additionally, we investigate the boundedness, periodicity, oscillationbehaviours, global asymptotically stability and rate of convergence of relatedhigher order difference equations.

Here, x=xx2+α⇒xx2+α=x⇒xx2+α-x=0⇒xx2+α-1=0⇒x=0, x2+α-1=0⇒x=0, x2=1-α⇒x=0, x=±1-α The solution of the…

Answered: (교)2 + a (7) + aT = T (피)3 + (a-1)교 =0…

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(교)2 + a (7) + aT = T (피)3 + (a-1)교 =0 F{(7) + (a – 1)} = 0 e equilibrium points are 0 and t/1- a. ||

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Explain the determine purple and inf is here and Why did we choose this equilibrium point?

Proposition [22] Assume that p,q € R. Then
\p| + |q] < 1
is a sufficient condition for the asymptotic stability of the difference equation
Хn+1 — рап — qxn-1 — 0, п 3 0, 1,... .
Lemma 1. The equilibrium points of the difference equation (2) are 0 and ±/1- a
Proof.
(x)
+ a
(7)° + ax = X
(7) + (a – 1) = 0
7{(T) + (a – 1)} = 0
This means that the equilibrium points are 0 and ±/1 – a.

Motivated by the above studies, we study the dynamics of following higher
order difference equation
Xn+1
A + B
(2)
п-т
where A, B are positive real numbers and the initial conditions are positive
numbers. Additionally, we investigate the boundedness, periodicity, oscillation
behaviours, global asymptotically stability and rate of convergence of related
higher order difference equations.

Expert Solution

Step 1

Here,

$x = \frac{x}{{(x)}^{2} + α} \Rightarrow x [{(x)}^{2} + α] = x \Rightarrow x [{(x)}^{2} + α] - x = 0 \Rightarrow x [{(x)}^{2} + (α - 1)] = 0 \Rightarrow x = 0, [{(x)}^{2} + (α - 1)] = 0 \Rightarrow x = 0, {(x)}^{2} = 1 - α \Rightarrow x = 0, x = \pm \sqrt{1 - α}$

The solution of the equation is: $x = 0, \pm \sqrt{1 - α}$

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