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

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Depending on the eigenvalues, the solutions have different behavior.

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Two Real nonzero eigenvalues. We have three cases:
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The two eigenvalues tex2html_wrap_inline44 and tex2html_wrap_inline46 are positive (with tex2html_wrap_inline48 ). When tex2html_wrap_inline50 , the solutions explode tangent to the straight-line solution associated to the eigenvalue tex2html_wrap_inline46 .

In this case the equilibrium point is a source.

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The two eigenvalues tex2html_wrap_inline44 and tex2html_wrap_inline46 are negative (with tex2html_wrap_inline60 ). When tex2html_wrap_inline50 , the solutions "die" at the origin. They tend to the equilibrium point tangent to the straight-line solution associated to the eigenvalue tex2html_wrap_inline44 .

In this case the equilibrium point is a sink.

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The two eigenvalues tex2html_wrap_inline44 and tex2html_wrap_inline46 have different signs (with tex2html_wrap_inline72 ). In this case, the solutions explode whether when tex2html_wrap_inline50 (except along the straight-line solution associated to the eigenvalue tex2html_wrap_inline44 ) or tex2html_wrap_inline78 (except along the straight-line solution associated to the eigenvalue tex2html_wrap_inline46 ).

In this case, the equilibrium point is a saddle.

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Repeated Real nonzero eigenvalue. Let us call this eigenvalue tex2html_wrap_inline84. We have two cases
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If tex2html_wrap_inline88, 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:

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If tex2html_wrap_inline92 , then the solutions get large as tex2html_wrap_inline50 . 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:

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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 tex2html_wrap_inline98 . Note that the solutions are all straight-line solutions. Depending on the sign of tex2html_wrap_inline84 , the solution may tend to or get away from the line of equilibrium points parallel to the eigenvector associated to the eigenvalue tex2html_wrap_inline84 . For a negative tex2html_wrap_inline84, we have

and for a positive tex2html_wrap_inline84, we have

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Complex eigenvalues. Let us write the eigenvalues as tex2html_wrap_inline106 . We have three cases.
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tex2html_wrap_inline110 . The solutions tend to the origin (when tex2html_wrap_inline50 ) while spiraling. In this case, the equilibrium point is called a spiral sink.

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tex2html_wrap_inline116 . The solutions explode or get away from the origin (when tex2html_wrap_inline50 ) while spiraling. In this case, the equilibrium point is called a spiral source.

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tex2html_wrap_inline122 . The solutions are periodic. This means that the trajectories are closed curves or cycles. In this case, the equilibrium point is called a center.

[Differential Equations] [First Order D.E.]
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