Theorem 4.1. The positive equilibrium point E, of system (4) is not asymptotically stableand nonhyperbolic point .Proof. The linearized system of (4) evaluated at Eo is written in the matrix form asLn+1Q Lnwhere Ln(Wn, Zn, Wn-1, Zn–1, Wn-2, zn-2)' and the Jacobian matrix determined at Eo is equal-2e-1-(e²+2e-3)-(4 µ)(e?+2 e-3)e+3e+3-(e+3)2 µ-(e+3)1-e2(28)1119. In this paper, we solve and study the properties of the following systemWn-p 2 Zn-hp=0Wn-ph=0E zn-hh=1p=1Wn+1+u and zn+1 =+e,(4)Zn - €Wn - 4where u and e are arbitrary positive real numbers with initial conditions w; and z; for i =-2, –1,0.Theorem 2.1. Let {wn, z„}-2 be a solution of (4), then6+ V – 4 ko k1A- V2 – 4 ko ki1- ki + ko+w2n-2+BTy – Oy -- V2 - 4 ko ki-4 ko ki+ V2 – 4 ko kiV2 - 4 ko kiAko+ Bkow2n-1- 122•((금-1) (는)1– ko + k1+ V2 - 4 vo V1- V2 - 4 VI1- v + vo22n-2+D%3D1+ Vu2 – 4 o VIgb + V2 – 4 vo vỊ11.+D- V2 – 4 v vỊV2 – 4 vo vi22n-1- v + vo1- 0 - VIwhere A, B, C and D are constants defined aswo -H2-1 +z-2koki =0 = ko + ki + ko ki,w-1 + w-220 - e20 - ew-1 +w-2, = vo + vi + vo V1,2-1 +2-21-OmVo2 – 4 ko k1-)") (v - 4ko ki – 4) + 2 (wo - (G ) -).2 62 - 8 ko kiV2- 4 ko ki262 - 8 ko ki1- ko + ki- ko – k1(1 – kj + ko1- k1 - koB =u- w-2+2wn-- vi +uC =-2+ 222 - 8 0 vI- o - VI) [(는1- v1 + o- v0 - v11- v + o2) (V2 – 4vo vI ++22 2-- 0 - v1since ko + ki #1 and vo + vı #1 for n e N.Proof. To obtain the expressions of the general solutions for (4), we rewrite it in the followform2 zn-hWn-pp=1Wn+1h=1Zn+1and(5)1E Wn-p1> Zn-hh=0wn -up-04Then, we assume thatWn+1 - lWn - Hkn+1 =1kn =2Wn-pp=0Wn-pp=1(6)Zn+1 -€Zn - €Vn+1 =Vn2E Zn-h> Zn-hh=1h=0Substituting (6) in (5), we haveZn-hh=11== kn-1,Vnkn+1 =(7)E Wn-pp=1Vn+1 == Vn-1.kn%3DWn -Hence, we see that1ki = -,Vowo - H20 - €kow-1+ w_2Vo =Z-1+ z-2koandat n = 1, k2 = ko andV2 = Vo,at n = 2, k3 = k1 andV3 = V1,at n = 3, k4 = ko andV4 = Vo,at n = 4, ks = k andV5 = V1,(8)at n = 2 n, k2n = ko andV2n = Vo,at n = 2n + 1, k2n+1 = k, andV2n+1 = V1.Now, from the relations in (6) we getWn = µ+ kn (wn-1 + wn-2) and zn = e + vn (žn-1 + zn-2).(9)Using (8) in (9), we haveW2n = H+ ko (w2n-1 + w2n-2),W2n+1 = µ+ k1 (w2n + w2n-1),(10)Z2n = €+ vo (z2n-1 + 2n-2),22n+1 = €+ V1 (22n + 22n-1).

Theorem 4.1. The positive equilibrium point Eo of system (4) is not asymptotically stable and nonhyperbolic point . Proof. The linearized system of (4) evaluated at Eo is written in the matrix form as Ln+1 = Q Ln,

Theorem 4.1. The positive equilibrium point Eo of system (4) is not asymptotically stable and nonhyperbolic point . Proof. The linearized system of (4) evaluated at Eo is written in the matrix form as Ln+1 = Q Ln,

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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