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Unformatted text preview: The phase portrait along with several graphs of x 1 versus t are given below. Behavior of Individual Trajectories As t → ∞ , each trajectory does one of the following: approaches infinity; approaches the critical point x = ; repeatedly traverses a closed curve, corresponding to a periodic solution, that surrounds the critical point. The trajectories never intersect, and exactly one trajectory passes through each point ( x , y ) in the phase plane. The only solution passing through the origin is x = . Other solutions may approach (0, 0), but never reach it. Behavior of Trajectory Sets As t → ∞ , one of the following cases holds: All trajectories approach the critical point x = . This is the case when the eigenvalues are real and negative or complex with negative real 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 = , 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. 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 = . Eigenvalues Type of Critical Point Stability 2 1 r r Node Unstable 2 1 < < r r Node Asymptotically Stable 1 2 r r < < Saddle Point Unstable 2 1 = r r Proper or Improper Node Unstable 2 1 < = r r Proper or Improper Node Asymptotically Stable μ λ i r r ± = 2 1 , Spiral Point λ Unstable < λ Asymptotically Stable μ μ i r i r = = 2 1 , Center Stable...
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 Spring '13
 MRR
 Math, Differential Equations, Linear Algebra, Equations, Critical Point, Linear Systems, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix, phase portraits, phase portrait

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