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Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 47EQ
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Show me the determine red and information is here

Xn-k (1) (æn-k)9-1 where the initial values x_kHI = a_k+i ,1 = 0,1, 2, ..k. are nonzero real numbers and with (x-k+1)9-1 + -a for l = 0,1,2, ..k. . Moreover , we have studied the stability and periodicity of solutions for the generalized nonlinear rational difference equations in (1) and for some special Xn+1 = + a cases . Xn Xn+1 = (2) (xn)² +a where (xo) + -a In order to do this we introduce the following notations: Let xo = a Proposition [22] Assume that p, g € R. Then |pl + lg| <1 is a sufficient condition for the asymptotic stability of the difference equation In+1 - prn - qan-1 = 0, n= 0,1, .... Lemma 1. The equilibrium points of the difference equation (2) are 0 and ±/1-a Proof. (7) +a (7) + az = 7 (포)3 + (a-1)표 =0 z{(2)? + (a – 1)} = 0 This means that the equilibrium points are 0 and ±V1-a. Remark 1. When a = 1, then the only equilibrium point of the difference equation (2) is 0. Theorem 7. 1) The equilibrium point z = 0 is locally asymptotically stable if Jal >1. 2)The equilibrium points 7 =tVT – aare locally asymptotically stable if 0< a <1. 3) All solutions of equation (2) at the equilibrium points 0 and ±/T – aare unstable if aso -13 Proof. let f: (0, 00) → (0, 00) be a continuous function defined by f(u) = u2 + a It is easy see that df (u) -u? +a du (u2 + a)? At the equilibrium point 7 = 0, we have df (u). \z=0 = du =p The corresponding linearized equation about 7 = 0 is given by Yn+1 - Pyn = 0 This implies that the characteristic equation is Hence the equilibrium point z= 0 is locally asymptotically stable if Ja| >1. Now we will prove the theorem at the equilibrium point = +V1-a and the proof at the equilibrium point 7= -V1- a by the same way. At the equilibrium point = v1- a we have df (u). -VEa = 2a - 1 =g du The corresponding linearized equation about 7 = V1- a is given by Yn+1 - qyn = 0 This implies that the characteristic equation is A- (2a – 1) = 0 Hence the equilibrium point E = VI- a is locally asymptotically stable if |2a – 1| <1. This means that 0< a < 1.
Transcribed Image Text:Xn-k (1) (æn-k)9-1 where the initial values x_kHI = a_k+i ,1 = 0,1, 2, ..k. are nonzero real numbers and with (x-k+1)9-1 + -a for l = 0,1,2, ..k. . Moreover , we have studied the stability and periodicity of solutions for the generalized nonlinear rational difference equations in (1) and for some special Xn+1 = + a cases . Xn Xn+1 = (2) (xn)² +a where (xo) + -a In order to do this we introduce the following notations: Let xo = a Proposition [22] Assume that p, g € R. Then |pl + lg| <1 is a sufficient condition for the asymptotic stability of the difference equation In+1 - prn - qan-1 = 0, n= 0,1, .... Lemma 1. The equilibrium points of the difference equation (2) are 0 and ±/1-a Proof. (7) +a (7) + az = 7 (포)3 + (a-1)표 =0 z{(2)? + (a – 1)} = 0 This means that the equilibrium points are 0 and ±V1-a. Remark 1. When a = 1, then the only equilibrium point of the difference equation (2) is 0. Theorem 7. 1) The equilibrium point z = 0 is locally asymptotically stable if Jal >1. 2)The equilibrium points 7 =tVT – aare locally asymptotically stable if 0< a <1. 3) All solutions of equation (2) at the equilibrium points 0 and ±/T – aare unstable if aso -13 Proof. let f: (0, 00) → (0, 00) be a continuous function defined by f(u) = u2 + a It is easy see that df (u) -u? +a du (u2 + a)? At the equilibrium point 7 = 0, we have df (u). \z=0 = du =p The corresponding linearized equation about 7 = 0 is given by Yn+1 - Pyn = 0 This implies that the characteristic equation is Hence the equilibrium point z= 0 is locally asymptotically stable if Ja| >1. Now we will prove the theorem at the equilibrium point = +V1-a and the proof at the equilibrium point 7= -V1- a by the same way. At the equilibrium point = v1- a we have df (u). -VEa = 2a - 1 =g du The corresponding linearized equation about 7 = V1- a is given by Yn+1 - qyn = 0 This implies that the characteristic equation is A- (2a – 1) = 0 Hence the equilibrium point E = VI- a is locally asymptotically stable if |2a – 1| <1. This means that 0< a < 1.
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