Classification in Two Dimensions
The roots of the characteristic polynomial det(A - λI) are the eigenvalues of A. The sign and relation of these roots, to each other may be used to determine the stability of the dynamical system
For a 2-dimensional system, the characteristic polynomial is of the form where is the trace and is the determinant of A. Thus the two roots are in the form:
Note also that and . Thus if then the eigenvalues are of opposite sign, and the fixed point is a saddle. If then the eigenvalues are of the same sign. Therefore if both are positive and the point is unstable, and if then both are negative and the point is stable. The discriminant will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).
Read more about this topic: Linear Dynamical System
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