9.3 Math 3331 Di erential Equations 9.3 Phase Plane Portraits Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math3331 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 1 / 24
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, 2014 2 / 24
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, 2014 3 / 24
Case A: T 2 4D > 0 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 4 / 24
Phase Portrait Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 5 / 24
Saddle: 1 > 0 > 2 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 6 / 24
Nodal Source: 1 > 2 > 0 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 7 / 24
Nodal Sink: 1 < 2 < 0 Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 8 / 24
Saddle: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 9 / 24
Nodal Source: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 10 / 24
Nodal Sink: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 11 / 24
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, 2014 12 / 24
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, 2014 13 / 24
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, 2014 14 / 24
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, 2014 15 / 24
Center: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 16 / 24
Spiral Source: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 17 / 24
Spiral Sink: Example Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 18 / 24
Degenerate Node: Borderline Case Spiral/Node Degenerate Node: Borderline Case Spiral/Node Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 19 / 24
Degenerate Nodal Source Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 20 / 24
Degenerate Nodal Sink Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 21 / 24
SaddleNode: Borderline Case Node/Saddle SaddleNode: Borderline Case Node/Saddle Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 22 / 24
Unstable Saddle-Node Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 23 / 24
Stable Saddle-Node Jiwen He, University of Houston Math 3331 Di erential Equations Summer, 2014 24 / 24