Notes-PhasePlane

# All the while it would roughly follow the 2 sets of

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changing direction and moves back to infinite-distant away. All the while it would roughly follow the 2 sets of eigenvectors. This type of critical point is called a saddle point . It is always unstable .

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© 2008 Zachary S Tseng D -2 - 7 Two distinct real eigenvalues, opposite signs Type: Saddle Point Stability: It is always unstable.
© 2008 Zachary S Tseng D -2 - 8 Case II. Repeated real eigenvalue 3. When there are two linearly independent eigenvectors k 1 and k 2 . The general solution is x = C 1 k 1 e rt + C 2 k 2 e rt = e rt ( C 1 k 1 + C 2 k 2 ) . Every nonzero solution traces a straight-line trajectory, in the direction given by the vector C 1 k 1 + C 2 k 2 . The phase portrait thus has a distinct star-burst shape. The trajectories either move directly away from the critical point to infinite-distant away (when r > 0), or move directly toward, and converge to the critical point (when r < 0). This type of critical point is called a proper node (or a starl point ). It is asymptotically stable if r < 0, unstable if r > 0. Note : For 2 × 2 systems of linear differential equations, this will occur if, and only if, when the coefficient matrix A is a constant multiple of the identity matrix: A = = α α α 0 0 1 0 0 1 , α = any nonzero constant * . * In the case of α = 0, the solution is = + = 2 1 2 1 1 0 0 1 C C C C x . Every solution is an equilibrium solution. Therefore, every trajectory on its phase portrait consists of a single point, and every point on the phase plane is a trajectory.

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© 2008 Zachary S Tseng D -2 - 9 A repeated real eigenvalue, two linearly independent eigenvectors Type: Proper Node (or Star Point) Stability: It is unstable if the eigenvalue is positive; asymptotically stable if the eigenvalue is negative.
© 2008 Zachary S Tseng D -2 - 10 4. When there is only one linearly independent eigenvector k . Then the general solution is x = C 1 k e rt + C 2 ( k t e rt + η e rt ). The phase portrait shares characteristics with that of a node. With only one eigenvector, it is a degenerated-looking node that is a cross between a node and a spiral point (see case 6 below). The trajectories either all diverge away from the critical point to infinite-distant away (when r > 0), or all converge to the critical point (when r < 0). This type of critical point is called an improper node . It is asymptotically stable if r < 0, unstable if r > 0.

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© 2008 Zachary S Tseng D -2 - 11 A repeated real eigenvalue, one linearly independent eigenvector Type: Improper Node Stability: It is unstable if the eigenvalue is positive; asymptotically stable if the eigenvalue is negative.
© 2008 Zachary S Tseng D -2 - 12 Case III. Complex conjugate eigenvalues The general solution is ( ) ( ) ) cos( ) sin( ) sin( ) cos( 2 1 t b t a e C t b t a e C x t t μ μ μ μ λ λ + + = 5. When the real part λ is zero. In this case the trajectories neither converge to the critical point nor move to infinite-distant away. Rather, they stay in constant, elliptical (or, rarely, circular) orbits. This type of critical point is called a center . It has a unique stability classification shared by no other: stable (or

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