{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

The phase portrait along with several graphs of x 1

Info iconThis preview shows pages 22–25. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Page22 / 25

The phase portrait along with several graphs of x 1 versus...

This preview shows document pages 22 - 25. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online