The origin is either a nodal or spiral sink all

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part. The origin is either a nodal or spiral sink. All trajectories remain bounded but do not approach the origin, and occurs when eigenvalues are pure imaginary. The origin is a center. Some trajectories, and possibly all trajectories except x = 0 , tend to infinity. This occurs when at least one of the eigenvalues is positive or has a positive real part. The origin is a nodal source, a spiral source, or a saddle point.
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Summary Table The following table summarizes the information we have derived about our 2 x 2 system x ' = Ax , as well as the stability of the equilibrium solution x = 0 . Eigenvalues Type of Critical Point Stability 0 2 1 r r Node Unstable 0 2 1 < < r r Node Asymptotically Stable 1 2 0 r r < < Saddle Point Unstable 0 2 1 = r r Proper or Improper Node Unstable 0 2 1 < = r r Proper or Improper Node Asymptotically Stable μ λ i r r ± = 2 1 , Spiral Point 0 λ Unstable 0 < λ Asymptotically Stable μ μ i r i r - = = 2 1 , Center Stable
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