Qualitative Analysis of Linear Systems

In this page, we will summarize the behavior of the solutions of linear systems. First consider the linear system

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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 straight-line 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 straight-line 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 straight-line solution associated to the eigenvalue

) or

(except along the straight-line 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

, then the solutions tend either to the equilibrium point tangent to the only straight-line solution,

or it can happen that all solutions (except for the equilibrium point) are straight-line solutions, approaching the equilibrium point:

, then the solutions get large as

. But even if the solution explodes, it does go to infinity either tangent to the straight-line 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 straight-line 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