2.3 General Phase Plane Analysis

Question 1

What are the two types of information?

Answer 1

Local and global

Question 2

How do we express second order ODEs?

Answer 2

As 2 couples first order ODEs

Question 3

What are autonomous and non-autonomous functions?

Answer 3

Autonomous: F = x dot and G = y dot are not explicitly t dependent
Non-autonomous: F and G are explicitly t dependent

Question 4

For any general function, express the forms of F, G and dy/dx

Answer 4

F(x,y) = dx/dt
G(x,y) = dy/dt
dy/dx = G/F

Question 5

Describe how the pendulum motion equation can be expressed using F and G

Answer 5

For a pendulum, x = Θ, dΘ/dt = y

F = y, G = -w^2 sinx

Question 6

Describe the important properties for dy/dx = G/F

Answer 6

• Uniquely defined tangent to y(x) everywhere except at F=G=0 ie x and y are fixed points
• Trajectories still cant cross unless at fixed points

Question 7

What are the two types of fixed points we see on a phase plane graph?

Answer 7

Repellors and attractors located on the x axis

Question 8

Describe what we see for conservative and non conservative cases for a Θ dot - Θ graph

Answer 8

See a circular orbit or a separatrix for the conservative case
See the spiral inwards and the inward pointing star for the non-conservative case - damping

Question 9

Describe the procedure for the phase plane analysis in the three steps

Answer 9

1. Solve F(x bar, y bar) = 0 and G = 0
2. Find dF/dx, dF/dy, dG/dx, dG/dy at x bar and y bar
3. p = dF/dx + dG/dy, q = dF/dx dG/dy - dF/dy dG/dx

Question 10

Describe the properties of an attractive star/spiral

Answer 10

Stable nodes, attractor points

Question 11

Describe the properties of an repulsive star/spiral

Answer 11

Unstable nodes, repellor points

Question 12

Are centre points (circles) and saddle points attractive or repulsive points?

Answer 12

They are neither