b) Show that the stability matrix (the Jacobian) for the linearization about this fixed point has the form -₁ -₂ -₂ d, -d, 0 0 0 -9-9₂ d, -d₁) We can write this in block-matrix form as 0 0 0 (A B BA where A and B are 2x2 matrices. Show that the four eigenvalues of the 4x4 block matrix are given by the eigenvalues of A-B and A+B.
b) Show that the stability matrix (the Jacobian) for the linearization about this fixed point has the form -₁ -₂ -₂ d, -d, 0 0 0 -9-9₂ d, -d₁) We can write this in block-matrix form as 0 0 0 (A B BA where A and B are 2x2 matrices. Show that the four eigenvalues of the 4x4 block matrix are given by the eigenvalues of A-B and A+B.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer letter B. Thank you.
![a) Show that x₁ * = y₁= x₂ * = y₂ = u is a fixed point for all choices of parame-
ters and that u is uniquely defined.
b) Show that the stability matrix (the Jacobian) for the linearization about this
fixed point has the form
-C₁-C₂-C3
0
d₁ -d₁ 0 0
-C3 0 -C₁-C₂
0
0 d₁ -d₁)
We can write this in block-matrix form as
(A B)
BA
where A and B are 2x2 matrices. Show that the four eigenvalues of the 4×4
block matrix are given by the eigenvalues of A-B and A+B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3dd9cd24-c83f-445a-b044-da2d154b184b%2F01cc1513-b4eb-4214-98ea-bb1b6f1d9a6c%2Frg0wcpj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a) Show that x₁ * = y₁= x₂ * = y₂ = u is a fixed point for all choices of parame-
ters and that u is uniquely defined.
b) Show that the stability matrix (the Jacobian) for the linearization about this
fixed point has the form
-C₁-C₂-C3
0
d₁ -d₁ 0 0
-C3 0 -C₁-C₂
0
0 d₁ -d₁)
We can write this in block-matrix form as
(A B)
BA
where A and B are 2x2 matrices. Show that the four eigenvalues of the 4×4
block matrix are given by the eigenvalues of A-B and A+B.
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