Phase Portraits for 2D Nonlinear Dynamical Systems - Homework 5 | MATH 341, Assignments of Mathematics

Download Phase Portraits for 2D Nonlinear Dynamical Systems - Homework 5 | MATH 341 and more Assignments Mathematics in PDF only on Docsity! Math 341 Dynamical Systems–Spring 2005–Pruett HW ASSIGNMENT 5 Chapter 6: Phase Portraits for 2D Nonlinear Dynamical Systems Counts as a 50pt HW Assignment ASSIGNED: Friday, March 18, 2005 DUE: Friday, March 25, 2005 1 6.1 Phase Portraits (10pts) Complete 6.1.5. Use different colors for nullclines and trajectories. 2 6.2 Existence and Uniqueness (10pts) Complete 6.2.2. 3 6.3 Fixed Points and Linearization (10pts) Complete 6.3.8. Note: the inverse-square law of gravitation (Newton) states that the attractive gravitational force between two objects separated by a distance r and of masses m1 and m2, respectively, is F = Gm1m2 r2 (1) where G is a universal (and positive) constant. (20pts) Complete 6.3.12 and also complete the analysis of Example 6.3.2 on p. 153. 1. (5pts) Consider the transformation between polar and rectangular coordinates: x(t) = r(t) cos θ(t) (2) y(t) = r(t) sin θ(t) (3) Show x2 + y2 = r2 and tan θ(t) = y(t) x(t) . 2. (5pts) Differentiate the equations above w.r.t. t to show that rṙ = xẋ + yẏ and (4) θ̇ = xẏ − yẋ r2 (5) 3. (5pts) Follow the steps on p. 153 to derive the following decoupled system in polar coordinates to replace the original system in rectangular coordinates (Ex. 6.3.2) ṙ = ar3 (6) θ̇ = 1 (7) 4. (5pts) In class, we showed that the original linearized system of Ex. 6.3.2 has borderline stability at the FP; that is, the FP is a center. Thus the stability of the nonlinear system will be determined by the nonlinear terms. Analyze the stability of Eq. 6 for a < 0, a = 0, and a > 0. Describe in words the behavior of the system in each case.