Linear Planar Systems Phase Line Portraits

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Last updated 7:58 PM on 5/9/26
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15 Terms

1
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λ(1) < 0 < λ(2) aka. different signs

saddle point

<p>saddle point</p>
2
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λ(1) < λ(2) < 0 aka both negative

nodal sink (whichever λ is more negative, goes quicker)

<p>nodal sink (whichever λ is more negative, goes quicker) </p>
3
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λ(1) > λ(2) > 0 aka both positive

nodal source

<p>nodal source </p>
4
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λ(1) < λ(2) = 0 aka one negative, one zero

spinal sink

<p>spinal sink </p>
5
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λ(1) > λ(2) = 0 aka one positive, one zero

spinal source

<p>spinal source</p>
6
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complex λ: if a21 < 0, what is the orientation?

CW

7
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complex λ: if a21 > 0, what is the orientation?

CCW

8
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complex λ: u + iv ; if u = 0

ellipse/center

<p>ellipse/center</p>
9
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complex λ: u + iv ; if u < 0

spiral sink

<p>spiral sink </p>
10
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complex λ: u + iv ; if u > 0

spiral source

<p>spiral source</p>
11
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Repeated λ (A = λI) s.t A is diagonalizable (A = λI) → λ < 0

radial sink

<p>radial sink </p>
12
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Repeated λ (A = λI) s.t A is diagonalizable (A = λI) → λ > 0

radial source

<p>radial source </p>
13
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Repeated λ (A = λI) s.t A is NOT diagonalizable (A = λI) → λ = 0

shear

<p>shear</p>
14
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Repeated λ (A = λI) s.t A is NOT diagonalizable (A = λI) → λ < 0

twist sink

<p>twist sink</p>
15
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Repeated λ (A = λI) s.t A is NOT diagonalizable (A = λI) → λ > 0

twist source

<p>twist source</p>