Saddle, Shrink, and Sprial

Denote , then for two variables and , assume the following first order equations: where the initial values is given. Then our interest is to find the eigenvalues of the following coefficient matrix:

Note that solving for the eigenvalues of the matrix is equal to finding the roots in the following equations: where and . Thus space is a first-order reformulation of the single-variable problem of . Solving for the eigenvalues is exactly the problem of solving for and .

Since the eigenvalues of the matrix is such that which equals to and by denoting we have the following combinations: