Qualitative Analysis of Linear Systems
In this page, we will summarize the behavior of the solutions of
linear systems. First consider the linear system
The associated eigenvalues are the roots of the characteristic polynomial
Depending on the eigenvalues, the solutions have different behavior.

 Two Real nonzero eigenvalues. We have three
cases:

 The two eigenvalues and are
positive (with ). When , the solutions explode tangent to the straightline solution
associated to the eigenvalue .
In this case the equilibrium point is a source.

 The two eigenvalues and are
negative (with ). When , the solutions "die" at the origin. They tend to
the equilibrium point tangent to the straightline
solution associated to the eigenvalue .
In this case the equilibrium point is a sink.

 The two eigenvalues and have
different signs (with ). In this case,
the solutions explode whether when (except
along the straightline solution associated to the eigenvalue
) or (except along the
straightline solution associated to the eigenvalue ).
In this case, the equilibrium point is a saddle.

 Repeated Real nonzero eigenvalue. Let us call
this eigenvalue .
We have two cases

 If ,
then the solutions tend either to the
equilibrium point tangent to the only straightline solution,
or it can happen that all solutions (except for the equilibrium point) are straightline solutions,
approaching the equilibrium point:

 If , then the solutions get large as . But even if the solution explodes, it does go
to infinity either tangent to the straightline solution,
or goes to infinity straight in every direction:

 Zero eigenvalue. If the system has zero as an
eigenvalue, then there exists a line of equilibrium points (degenerate
case). Let us call the other eigenvalue . Note that
the solutions are all straightline solutions. Depending on the sign
of , the solution may tend to or get away from the line of
equilibrium points parallel to the eigenvector associated to the
eigenvalue . For a negative , we have
and for a positive , we have

 Complex eigenvalues. Let us write the
eigenvalues as . We have three cases.

 . The
solutions tend to the origin (when ) while
spiraling. In this case, the equilibrium point is called a
spiral sink.

 . The
solutions explode or get away from the origin (when ) while spiraling. In this case, the equilibrium point is
called a spiral source.

 . The
solutions are periodic. This means that the trajectories are closed
curves or cycles. In this case, the equilibrium point is
called a center.
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Author: Mohamed
Amine Khamsi
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