Di erential Equations

Transcription

1 9.3 Math 3331 Di erential Equations 9.3 Phase Plane Portraits Jiwen He Department of Mathematics, University of Houston math.uh.edu/ jiwenhe/math3331 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

2 9.3 Phase Plane Portraits Classification of 2d Systems Distinct Real Eigenvalues Phase Portrait Saddle: 1 > 0 > 2 Nodal Source: 1 > 2 > 0 Nodal Sink: 1 < 2 < 0 Complex Eigenvalues Center: =0 Spiral Source: >0 Spiral Sink: <0 Borderline Cases Degenerate Node: Borderline Case Spiral/Node Degenerate Nodal Source Degenerate Nodal Sink SaddleNode: Borderline Case Node/Saddle Unstable Saddle-Node Stable Saddle-Node Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

3 Classification of 2d Systems Case B: T 2 4D < 0 ) complex eigenvalues 1,2 = ± i = T /2, = p 4D T 2 /2 Case C: T 2 4D =0 ) single eigenvalue = T /2 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

4 Case A: T 2 4D > 0 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

5 Phase Portrait Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

6 Saddle: 1 > 0 > 2 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

7 Nodal Source: 1 > 2 > 0 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

8 Nodal Sink: 1 < 2 < 0 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

9 Saddle: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

10 Nodal Source: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

11 Nodal Sink: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

12 Case A: T 2 4D < 0 Case B: T 2 4D < 0 ) complex eigenvalues 1,2 = ± i = T /2, = p 4D T 2 /2 complex ) eigenvector v = u + iw complex ) no half line solutions General solution: x(t) =e at applec 1 (u cos t w sin t) Subcases of Case B Center: =0 Spiral Source: >0 Spiral Sink: <0 Borderline Case: Center ( = 0) is border between spiral source ( >0) and spiral sink ( <0). + c 2 (u sin t + w cos t) Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

13 Center: =0 Direction of Rotation: At x =[1, 0] T, y 0 = c. If c > 0, ) counterclockwise, If c < 0, ) clockwise. Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

14 Spiral Source: >0 Direction of Rotation: At x =[1, 0] T, y 0 = c. If c > 0, ) counterclockwise, If c < 0, ) clockwise. Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

15 Spiral Sink: <0 Direction of Rotation: At x =[1, 0] T, y 0 = c. If c > 0, ) counterclockwise, If c < 0, ) clockwise. Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

16 Center: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

17 Spiral Source: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

18 Spiral Sink: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

19 Degenerate Node: Borderline Case Spiral/Node Degenerate Node: Borderline Case Spiral/Node Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

20 Degenerate Nodal Source Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

21 Degenerate Nodal Sink Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

22 SaddleNode: Borderline Case Node/Saddle SaddleNode: Borderline Case Node/Saddle Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

23 Unstable Saddle-Node Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24

24 Stable Saddle-Node Jiwen He, University of Houston Math 3331 Di erential Equations Summer, / 24