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Hence we obtain the linearized equation of Eq-(8) about its unique positive equilibrium point y as follow: 2p Zn+1 Zn + Zn-m = 0. (15) Therefore, the characteristic equation of Eq.(8) is 2p = 0. + uY y? m+1 (16)

Hence we obtain the linearized equation of Eq-(8) about its unique positive equilibrium point y as follow: 2p Zn+1 Zn + Zn-m = 0. (15) Therefore, the characteristic equation of Eq.(8) is 2p = 0. + uY y? m+1 (16)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Motivated by the above studies, we study the dynamics of following higher order difference equation Xn Xn+1 = A+ B- 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 Yn+1 = 1+p (8) Yn-m where p = . From now on, we handle the difference equation (8). The unique positive equilibrium point of Eq.(8) is 1+ V1+ 4p
Transcribed Image Text:Motivated by the above studies, we study the dynamics of following higher order difference equation Xn Xn+1 = A+ B- 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 Yn+1 = 1+p (8) Yn-m where p = . From now on, we handle the difference equation (8). The unique positive equilibrium point of Eq.(8) is 1+ V1+ 4p
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 I be a continuously differentiable function such that f is defined by Уп -m) = 1+p2 Уп-т f (Yn, Yn-1,', Yn-m, Therefore we have af 9o = dyn q1 = 42 = ·…:= qm-1 = 0, af dyn- 2p Im = 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) Therefore, the characteristic equation of Eq.(8) is Am+1 2p = 0. (16)
Transcribed Image Text: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 I be a continuously differentiable function such that f is defined by Уп -m) = 1+p2 Уп-т f (Yn, Yn-1,', Yn-m, Therefore we have af 9o = dyn q1 = 42 = ·…:= qm-1 = 0, af dyn- 2p Im = 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) Therefore, the characteristic equation of Eq.(8) is Am+1 2p = 0. (16)
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