Compartment Graphs and Linear Systems - Environmental Modeling - page 333

Table 18.2 Classification of equilibria stability according to eigenvectors, for N ¼ 2

Matrix of equivalent
reference system

stable (s)
unstable (u) Comment Eigenvalues l 1 , l 2

Re ð l 1 Þ< 0 ; Re ð l 2 Þ< 0 s Hyperbolic sink

10
0 1

u

Re ð l 1 Þ> 0 ; Re ð l 2 Þ> 0 Hyperbolic source

10
01

Re ð l 1 Þ> 0 ; Re ð l 2 Þ< 0 u Hyperbolic saddle

10
0 1

Re ð l 1 Þ< 0 ; l 2 ¼ 0 s Equilibria for all c 2 , c 1 ¼ 0

10
00

u

Re ð l Þ> 0 ; l ¼ 0 Equilibria for all c 2 , c 1 ¼ 0

10
00

1 2

s

l 1 ¼ l 2 ¼ 0 2 eigenvectors 00
00

All locations equilibria

u

l 1 ¼ l 2 ¼ 0 1 eigenvector 01

00

Equilibria for all c 1 , c 2 ¼ 0

Re ð l Þ¼ Re ð l Þ¼ 0 s Oscillations

01
10

1 2

1

variable 2

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

variable 1

-1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Fig. 18.9 Phase diagram for the reference matrix of two purely imaginary eigenvalues,
representing oscillations around the equilibrium; obtained using phasediag.m file with reference
matrix from Table 18.2 , zero right hand side and equidistant start positions on the c 2 -axis