Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Now, we consider a transformation as follows: (In, In-1, Zn, Zn-1) → (f, fi, 8, g1) where f = 1+ p, fi = tn, 8 =1+ p, 81 = zn. Thus we get the jacobian matrix about equilibrium point (7, 2): %3D

Now, we consider a transformation as follows: (In, In-1, Zn, Zn-1) → (f, fi, 8, g1) where f = 1+ p, fi = tn, 8 =1+ p, 81 = zn. Thus we get the jacobian matrix about equilibrium point (7, 2): %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Show me the steps of deremine yellow and inf is here i need evey I need all the details step by step and inf is here

Motivated by difference equations and their systems, we consider the following system of difference equations Ул Xn Xn+1 = A + B- 2 , Yn+1 = A + B- 2 (1) Yn-1 where A and B are positive numbers and the initial values are positive numbers. In First of all, we consider the change of the variables for system (1) as follows: Xn tn = Уп , Zn = A A From this, system (1) transform into following system: tn Zn Zn+1 = 1+ p n-1 tn+1 = 1+P2 (2) n-1 where P = > 0. From now on, we study the system (2). Lemma 1 Let p > 0. Unique positive equilibrium point of system (2) is (1+ /I+4p 1+ VI+4p (f,३) = 2 Now, we sider a transfo as follows: (In, In-1, Zn, Zn-1) → (f, f1, g, g1) where f = 1+ p, fi = tn, 8 = 1+ p, 81 = Zn. Thus we get the jacobian matrix about equilibrium point (7, z): n-I 'n-1 0 0 1 B (7,३) = 0 0 1 Thus, the linearized system of system (2) about the unique positive equilibrium point is given by XN+1 = B (i, z) X N, where In tn-1 XN = uz Zn-1 0 0 울글 1 0 0 B (i, 2): -21 0 0 1 Hence, the characteristic equation of B (î, z) about the unique positive equilibrium point (7, 2) is 4p? 4p2 = 0. Due to t = 7, we can rearrange the characteristic equation such that 4p2 2²+ 74 4p2 = 0. 14 Therefore, we obtain the four roots of characteristic equation as follows: p+ Vp? – 8pi? 212 A1 = Vp² – 8 pi? 272 P-VP- 12 = -p+Vp? + 8pi? 212 13 =
Transcribed Image Text:Motivated by difference equations and their systems, we consider the following system of difference equations Ул Xn Xn+1 = A + B- 2 , Yn+1 = A + B- 2 (1) Yn-1 where A and B are positive numbers and the initial values are positive numbers. In First of all, we consider the change of the variables for system (1) as follows: Xn tn = Уп , Zn = A A From this, system (1) transform into following system: tn Zn Zn+1 = 1+ p n-1 tn+1 = 1+P2 (2) n-1 where P = > 0. From now on, we study the system (2). Lemma 1 Let p > 0. Unique positive equilibrium point of system (2) is (1+ /I+4p 1+ VI+4p (f,३) = 2 Now, we sider a transfo as follows: (In, In-1, Zn, Zn-1) → (f, f1, g, g1) where f = 1+ p, fi = tn, 8 = 1+ p, 81 = Zn. Thus we get the jacobian matrix about equilibrium point (7, z): n-I 'n-1 0 0 1 B (7,३) = 0 0 1 Thus, the linearized system of system (2) about the unique positive equilibrium point is given by XN+1 = B (i, z) X N, where In tn-1 XN = uz Zn-1 0 0 울글 1 0 0 B (i, 2): -21 0 0 1 Hence, the characteristic equation of B (î, z) about the unique positive equilibrium point (7, 2) is 4p? 4p2 = 0. Due to t = 7, we can rearrange the characteristic equation such that 4p2 2²+ 74 4p2 = 0. 14 Therefore, we obtain the four roots of characteristic equation as follows: p+ Vp? – 8pi? 212 A1 = Vp² – 8 pi? 272 P-VP- 12 = -p+Vp? + 8pi? 212 13 =
Expert Solution
steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS