This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 404  Homework 6 Example Student Solutions February 15, 2011 S: 6.1.8 Plot Phase Portraits for the following system (the van der Pol oscilla tor). x = y y = x + y (1 x 2 ) Answer First, note that we will have fixed points when x = 0 and y = 0. This will happen when: 0 = y 0 = x + 0 x = 0 Therefore, there is only one fixed point and it occurs at (0 , 0). We will thus focus our phase portrait graph around this point. Here we see that there is a closed orbit around the fixed point, and that within that closed orbit the fixed point acts like an unstable spiral. 1 Figure 1: Phase Portrait 2 S: 6.1.9 Plot Phase Portraits for the following system (the dipole fixed point). x = 2 xy y = y 2 x 2 Answer First note that we will have fixed points when x = 0 and y = 0. This will happen when: 0 = 2 xy and 0 = y 2 x 2 The second equation tells us that y = x and thus the first one implies that this only happens at (0 , 0). Therefore, there is only one fixed point, and we will focus our phase portrait around this point. Figure 2: Phase Portrait 3 S: 6.1.10 Plot Phase Portraits for the following system (the twoeyed mon ster). x = y + y 2 y = 1 2 x + 1 5 y xy + 6 5 y 2 Answer First note that we will have fixed points when x = 0 and y = 0. 0 = y + y 2 factros to give us ( y + 1) y = 0 and finally y = 1 and y = 0 When y = 0 and y = 0, we know that x = 0. Therefore, there is a fixed point at (0 , 0). When y = 1 and y = 0: 0 = 1 2 x 1 5 + x + 6 5 1 = 1 2 x x = 2 Therefore, there is another fixed point at ( 2 , 1). We will thus focus the phase portrait around these two points....
View
Full
Document
This note was uploaded on 03/13/2011 for the course MATH 404 taught by Professor Staff during the Spring '08 term at University of Michigan.
 Spring '08
 STAFF
 Math, Algebra

Click to edit the document details