Lecture 29: Phase Plane Portraits for Planar Systems with Distinct Eigenvalues - Prof. Jiw, Study notes of Differential Equations

Download Lecture 29: Phase Plane Portraits for Planar Systems with Distinct Eigenvalues - Prof. Jiw and more Study notes Differential Equations in PDF only on Docsity! Section 9.2 Section 9.3 Lecture 29 9.2 Planar Systems 9.3 Phase Plane Portraits (I) Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/∼jiwenhe/math3331 Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 1 / 22 Section 9.2 Section 9.3 2D Distinct Eigenvalues In-Class Exercises Planar Systems Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 2 / 22 Section 9.2 Section 9.3 2D Distinct Eigenvalues In-Class Exercises Exercise 9.2.3 Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 5 / 22 Section 9.2 Section 9.3 2D Distinct Eigenvalues In-Class Exercises Exercise 9.2.9 Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 6 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Classification of 2d Systems Case B: T 2 − 4D < 0 ⇒ complex eigenvalues λ1,2 = α± iβ α = T/2, β = √ 4D − T 2/2 Case C: T 2 − 4D = 0 ⇒ single eigenvalue λ = T/2 Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 7 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Saddle: λ1 > 0 > λ2 Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 10 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Nodal Source: λ1 > λ2 > 0 Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 11 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Nodal Sink: λ1 < λ2 < 0 Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 12 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Nodal Sink: Example Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 15 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Case A: T 2 − 4D < 0 Case B: T 2 − 4D < 0 ⇒ complex eigenvalues λ1,2 = α± iβ α = T/2, β = √ 4D − T 2/2 λ complex ⇒ eigenvector v = u + iw complex ⇒ no half line solutions General solution: x(t) = eat [ c1(u cos βt −w sin βt) + c2(u sin βt + w cos βt) ] Subcases of Case B Center: α = 0 Spiral Source: α > 0 Spiral Sink: α < 0 Borderline Case: Center (α = 0) is border between spiral source (α > 0) and spiral sink (α < 0). Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 16 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Center: α = 0 Direction of Rotation: At x = [1, 0]T , y ′ = c . If c > 0, ⇒ counterclockwise, If c < 0, ⇒ clockwise. Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 17 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Center: Example Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 20 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Spiral Source: Example Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 21 / 22 Section 9.2 Section 9.3 Distinct Eigenvalues Complex Eigenvalues Spiral Sink: Example Jiwen He, University of Houston Math 3331 – Section 19470, Lecture 29 April 15, 2009 22 / 22