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Theorem 8 Let p ≤ (0, ³). Then the equilibrium point ã of Eq.(8) is locally asymptotically stable. Proof. From (15), we have Note that Thus |91| + |92|+... ·|9m| = Р Hence, we get from p > 0, 191 +92 +9m| 3 (2p+1-√4p+ 2p 2p+1-√4p+1 2p 4p+3-3√4p + 1 2p = 3p < 1, 3p < 0. 4p- 1) (√4p+1 C(√4P+1-1 So, we obtain 0 < p < . Therefore, the proof of Theorem 8 is completed. <1, / -2)< 0.

Theorem 8 Let p ≤ (0, ³). Then the equilibrium point ã of Eq.(8) is locally asymptotically stable. Proof. From (15), we have Note that Thus |91| + |92|+... ·|9m| = Р Hence, we get from p > 0, 191 +92 +9m| 3 (2p+1-√4p+ 2p 2p+1-√4p+1 2p 4p+3-3√4p + 1 2p = 3p < 1, 3p < 0. 4p- 1) (√4p+1 C(√4P+1-1 So, we obtain 0 < p < . Therefore, the proof of Theorem 8 is completed. <1, / -2)< 0.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Theorem 8 Let p e (0, ). Then the equilibrium point i of Eq. (8) is locally asymptotically stable. Proof. From (15), we have lg1|+ |92|+ am| Note that 2p +1- V4p +1 2p Thus 3p < 1, y2 l91|+ 192| + 3 (2p +1- V4p+ 1) < 1, 2p 4p + 3 — З/4p+1 2p < 0. Hence, we get from p > 0, 4p + 1 – 1) (V4p +1– 2) < 0. So, we obtain 0 <p< . Therefore, the proof of Theorem 8 is completed. I
Transcribed Image Text:Theorem 8 Let p e (0, ). Then the equilibrium point i of Eq. (8) is locally asymptotically stable. Proof. From (15), we have lg1|+ |92|+ am| Note that 2p +1- V4p +1 2p Thus 3p < 1, y2 l91|+ 192| + 3 (2p +1- V4p+ 1) < 1, 2p 4p + 3 — З/4p+1 2p < 0. Hence, we get from p > 0, 4p + 1 – 1) (V4p +1– 2) < 0. So, we obtain 0 <p< . Therefore, the proof of Theorem 8 is completed. I
Yn+1 = 1+P-m 4 Stability of Eq.(8) In this here, we study the stability of Eq.(8). Firstly we handle the linearized equation of Eq.(8) about its unique positive equilibrium point. Let I be some interval of real numbers and let f : Im+1 be a continuously differentiable function such that f is defined by Yn f (yn, Yn-1,* , Yn-m) = 1+p Уп-т Therefore we have af qo = Əyn %3D q1 = q2 ...= qm-1= 0, af Əyn-m 2p Im - Hence we obtain the linearized equation of Eq.(8) about its unique positive equilibrium point ỹ as follow: 2p 2n-m = 0. (15) Zn+1 Zn + Therefore, the characteristic equation of Eq.(8) is 2p m+1 (16) | Hence we obtain the linearized equation of Eq.(8) about its unique positive equilibrium point j as follow: 2p Zn+1 Zn + Zn-m = 0. (15) Motivated by the above studies, we study the dynamics of following higher order difference equation In+1 In A+B- (2) %3D "n-m 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. Firstly, we take the change of the variables for Eq.(2) as follows yn = From this, we obtain the following difference equation Yn .2 (8) where p = . From now on, we handle the difference equation (8). The unique positive equilibrium point of Eq.(8) is 1+ VI+ 4p
Transcribed Image Text:Yn+1 = 1+P-m 4 Stability of Eq.(8) In this here, we study the stability of Eq.(8). Firstly we handle the linearized equation of Eq.(8) about its unique positive equilibrium point. Let I be some interval of real numbers and let f : Im+1 be a continuously differentiable function such that f is defined by Yn f (yn, Yn-1,* , Yn-m) = 1+p Уп-т Therefore we have af qo = Əyn %3D q1 = q2 ...= qm-1= 0, af Əyn-m 2p Im - Hence we obtain the linearized equation of Eq.(8) about its unique positive equilibrium point ỹ as follow: 2p 2n-m = 0. (15) Zn+1 Zn + Therefore, the characteristic equation of Eq.(8) is 2p m+1 (16) | Hence we obtain the linearized equation of Eq.(8) about its unique positive equilibrium point j as follow: 2p Zn+1 Zn + Zn-m = 0. (15) Motivated by the above studies, we study the dynamics of following higher order difference equation In+1 In A+B- (2) %3D "n-m 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. Firstly, we take the change of the variables for Eq.(2) as follows yn = From this, we obtain the following difference equation Yn .2 (8) where p = . From now on, we handle the difference equation (8). The unique positive equilibrium point of Eq.(8) is 1+ VI+ 4p
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