Lecture 14: approaching chaos

Question 1

Dimension of phase space for a damped, driven pendulum

Answer 1

3

Question 2

Euler-Cromer

Question 3

Line attractor

Answer 3

for a damped, driven oscillation, we reach a steady state in which the phase space shows only an ellipse-like trajectory

Question 4

Point attractor

Answer 4

all trajectories decay to (theta, theta dot) = (0,0) as t appraoches infinity, for an undriven, damped oscillator

Question 5

Single-period limit cycle

Answer 5

At the damping coefficient of the system, after every period, the motion completes one full ellipse

Question 6

Poincare section

Answer 6

Where you plot one point every forcing period

Question 7

Subharmonic

Question 8

Two-period attractor

Answer 8

you bounce between two cycles. the first period is the time it takes to get from one dot to the other dot. the second period is the time it takes to get from one dot to the other dot, then back to the first dot

Question 9

Basin of attraction

Answer 9

Occurs in 3D with high enough non-linearity

Question 10

Why can 3D trajectories lead to deterministic chaos?

Answer 10

They can form complicated patterns

Question 11

By changing the initial condition, how does this affect the basin of attraction?

Answer 11

We change the basin of attraction we start in

Question 11

From the phase plot, how can you tell whether chaos has occurred?

Answer 11

The signal doesn’t appear periodic anymore

Question 12

At small driving amplitudes, oscillators end up along what type of attractors in phase space?

Answer 12

Line attractors