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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
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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