evaluating locally linear systems (stability, type, phase portrait)

Description

We want to solve x' = Ax. If detA =/= 0, then the origin is the only critical point. The following are different classifications of the zero vector for type and stability with corresponding phase portraits.
Georgie D'Sanson
Flashcards by Georgie D'Sanson, updated more than 1 year ago
Georgie D'Sanson
Created by Georgie D'Sanson over 4 years ago
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Question Answer
r1 < 0 < r2 (real and distinct eigenvalues r1, r2)
0 < r1 < r2 (real and distinct eigenvalues r1, r2) nodal source
r1 < r2 < 0 (real and distinct eigenvalues r1, r2) nodal sink
λ = 0 (r1,r2 are complex conjugates r1 = λ + iμ)
λ > 0 (r1,r2 are complex conjugates r1 = λ + iμ) spiral source
λ < 0 (r1,r2 are complex conjugates r1 = λ + iμ) spiral sink
r > 0 (r1 = r2, 1 linearly independent eigenvector) (source)
r < 0 (r1 = r2, 1 linearly independent eigenvector) (sink)
r > 0 (r1 = r2, 2 linearly independent eigenvectors) star node (source) unstable
r < 0 (r1 = r2, 2 linearly independent eigenvectors) star node (sink) asymptotically stable
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