Topological analysis of vector fields
The idea of this technique is to investigate critical points of a vector field, based on the eigenvalues of the Jacobian J at these critical points. A critical point is defined as an isolated point where the vector field vanishes, i.e., where u = 0. The following cases can be identified, based on the eigenvalues λ1 and λ2 of the Jacobian:
Trace[J] = a + d ( a, d are components of matrix J)
(Trace[J])^2 = 4 det[𝐽] is parabola
Attracting node: The eigenvalues are real and negative, corresponding to a sink.
(Trace[J])^2 − 4 det[𝐽] = 0, 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑝𝑎𝑟𝑡 = 0, Trace[J]<0, det[J]>0
Repelling node: The eigenvalues are real and positive, corresponding to a source.
(Trace[J])^2 − 4 det[𝐽] = 0, 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑝𝑎𝑟𝑡 = 0, Trace[J]>0, det[J]>0
Saddle point: The eigenvalues are real and have opposite sign.
(Trace[J])^2 − 4 det[𝐽] = 0, 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑝𝑎𝑟𝑡 = 0, det[J]<0 Trace[J]<0 or Trace[J]<0 (i.e a+d는 무관)
Repelling focus: Both eigenvalues are complex conjugate and their real part is positive.
(Trace[J])7 − 4 det[𝐽] ≠ 0, 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑝𝑎𝑟𝑡 ≠ 0, Trace[J]>0
Attracting focus: Both eigenvalues are complex conjugate and their real part is negative.
(Trace[J])7 − 4 det[𝐽] ≠ 0, 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑝𝑎𝑟𝑡 ≠ 0, Trace[J]<0 ,
Parameter set 1
critical point (0, 0)
Jacobian
Det |J-λI| = 0, λ1=1, λ2=-1
The eigenvalues are real and have opposite sign.
ð Saddle point.
Parameter set 2
critical point (0, 0)
Jacobian
Det |J-λI| = 0, λ=1
The eigenvalue is real and positive
ð Repelling node.
Parameter set 3
critical point (0, 0)
Jacobian
Det |J-λI| = 0, λ=-1
The eigenvalue is real and negative.
ð Attracting node.
Parameter set 4
critical point (0, 0)
Jacobian
Det |J-λI| = 0, (-2-λ)^2+1=0 , λ^2+4 λ+5=0 , 
,
Both eigenvalues are complex conjugate and their real part is negative.
ð Attracting focus.
The source of the problem is Professor Filip Sadlo in University of Heidelberg
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