005 - 5.1 Equlllbdtnn Point Analysis 3Zl(f(3 Solutions tend...

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Unformatted text preview: 5.1 Equlllbdtnn Point Analysis 3Zl (f) (3) Solutions tend away from the origin along the y-axis in both systems. In the nonlinear system, solutions approach the origin along the curve y = x3 which is tangent to the x- axis. For the linearized system. solutions tend to the origin along the x-axis. Near the origin, the phase portraits are almost the same. 6. Warning: 'lhe system stated in the exercise in the first two printings has a typo. The coefficient to the xy-term for dy/dt should be -2, not - I. This answer assumes that that coefficient is —2. (a) The Jacobian is 2 — 2x - y —x —2y 3 — 2y — 2x ‘ Evaluating at (0. 0). we get the linearized system dx 3 — 2x dy 2; — 3y. Evaluating at (0. 3), we get ‘2 — -1. d: — dy E; - "GX - 3y o and evaluating at (2, 0). we get dx _ = -2x _ dz 2y ‘2 _ dt - (b) At (0, 0), the eigenvalues are 2 and 3, so (0, 0) is a source. At (0, 3), eigenvalues are —l and —3, so (0. 3) is a sink. At (2, 0). the eigenvalues are —2 and — l . so (2. 0) is a sink. ...
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