Systems with Zero as an Eigenvalue

We discussed the case of system with two distinct real eigenvalues, repeated (nonzero) eigenvalue, and complex eigenvalues. But we did not discuss the case when one of the eigenvalues is zero. In fact, it is easy to see that this happen if and only if we have more than one equilibrium point (which is (0,0)). In this case, we will have a line of equilibrium points (the direction vector for this line is the eigenvector associated to the eigenvalue zero).

Example. Find the general solution to

displaymath64

Answer. The characteristic polynomial of this system is

displaymath66

which reduces to tex2html_wrap_inline68 . The eigenvalues are tex2html_wrap_inline70 and tex2html_wrap_inline72 . Let us find the associated eigenvectors.

tex2html_wrap_inline74
For tex2html_wrap_inline70 , set

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The equation tex2html_wrap_inline80 translates into

displaymath82

The two equations are the same. So we have y = 2x. Hence an eigenvector is

displaymath86

tex2html_wrap_inline74
For tex2html_wrap_inline72 , set

displaymath92

The equation tex2html_wrap_inline94 translates into

displaymath96

The two equations are the same (as -x-y=0). So we have y = -x. Hence an eigenvector is

displaymath102

Therefore the general solution is

displaymath104

Note that all the solutions are line parallel to the vector tex2html_wrap_inline106 . When tex2html_wrap_inline108 , the trajectory goes to infinity. But when tex2html_wrap_inline110 , the trajectory converge to the equilibrium point on the line of equilibrium points (that is passing by (0,0) and having tex2html_wrap_inline114 as a direction vector). The picture below explains more what is happening.

The general case is very similar to this example. Indeed, assume that a system has 0 and tex2html_wrap_inline116 as eigenvalues. Hence if tex2html_wrap_inline114 is an eigenvector associated to 0 and tex2html_wrap_inline120 an eigenvector associated to tex2html_wrap_inline122 , then the general solution is

displaymath124

We have two cases, whether tex2html_wrap_inline126 or tex2html_wrap_inline128 .

tex2html_wrap_inline74
If tex2html_wrap_inline132 , then tex2html_wrap_inline134 is an equilibrium point.
tex2html_wrap_inline74
If tex2html_wrap_inline138 , then the solution is a line parallel to the vector tex2html_wrap_inline120 . Moreover, we have when tex2html_wrap_inline108
tex2html_wrap_inline74
if tex2html_wrap_inline126 , the solution tends away from the line of equilibrium;

tex2html_wrap_inline74
if tex2html_wrap_inline128 , the solution tends to the equilibrium point tex2html_wrap_inline152 along a line parallel to tex2html_wrap_inline120 .

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Author: Mohamed Amine Khamsi

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