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It has a repeated eigenvalue r = − 1 that has only

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Unformatted text preview: It has a repeated eigenvalue r = − 1 that has only one linearly independent eigenvector. Therefore, the critical point at (3, 1) is an asymptotically stable improper node . The phase portrait is shown below. © 2008 Zachary S Tseng D-2 - 19 Example : x 1 ′ = −2 x 1 − 6 x 2 + 8 x 2 ′ = 8 x 1 + 4 x 2 − 12 The critical point is at (1, 1). The matrix A has characteristic equation r 2 − 2 r + 40 = 0. It has complex conjugate eigenvalues with positive real part, λ = 1. Therefore, the critical point at (1, 1) is an unstable spiral point . The phase portrait is shown below. © 2008 Zachary S Tseng D-2 - 20 Exercises D-2.1 : 1 – 8 Determine the type and stability or the critical point at (0,0) of each system below. 1. x ′ = − − 10 5 7 2 x . 2. x ′ = − − 3 3 6 3 x . 3. x ′ = − 4 1 4 8 x . 4. x ′ = 6 2 8 6 x . 5. x ′ = − − − 1 1 1 1 x . 6. x ′ = − − 4 4 x . 7. x ′ = 3 2 4 1 x . 8. x ′ = − 6 5 6 x . 9. (i) For what value(s) of b will the system below have an improper node at (0,0)? (ii) For what value(s) of b will the system below have a spiral point at (0,0)? x ′ = − 1 2 5 b x . 10 – 14 Find the critical point of each nonhomogeneous linear system given. Then determine the type and stability of the critical point. 10. x ′ = − 3 6 2 1 x + − 24 6 . 11. x ′ = 3 3 x + − 3 12 . 12. x ′ = − 7 6 x + 7 3 . 13. x ′ = − − 1 2 2 3 x + − 2 4 . 14. x ′ = − − 1 3 6 5 x + − 13 . © 2008 Zachary S Tseng D-2 - 21 Answers D-2.1 : 1. Node, asymptotically stable 2. Center, stable 3. Improper node, unstable 4. Node, unstable 5. Spiral point, asymptotically stable 6. Proper node (or start point), asymptotically stable 7. Saddle point, unstable 8. Improper node, unstable 9. (i) b = 9/2, (ii) b < −9/2 10. (−2, 4) is a saddle point, unstable. 11. (−4, 1) is a proper node (star point), unstable. 12. (1, −0.5) is a center, stable. 13. (0, 2) is an improper node, asymptotically stable 14. (−1, −3) is a spiral point, unstable. © 2008 Zachary S Tseng D-2 - 22 Nonlinear Systems Consider a nonlinear system of differential equations: x ′ = F ( x , y ) y ′ = G ( x , y ) Where F and G are functions of two variables: x = x ( t ) and y = y ( t ); and such that F and G are not both linear functions of x and y ....
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It has a repeated eigenvalue r = − 1 that has only one...

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