Dynamic Systems Assignment

 

1)   Graphing one finger motion.

a)    The paradigm.

i)     Hold your right hand out in front of you with only your index finger extended.

ii)   Put the back of your hand towards the ceiling.

iii) Move your finger left and right.

b)   Graph this finger movement.

i)     Put time on the x-axis.

(1) Let time range from 0 to 10 seconds.

ii)   Put finger position on the y-axis.

(1) If the finger is all the way to the outside of your body (away from your other hand or towards your pinky), finger position in the graph is high (up in the graph).

(2) If the finger is all the way to the center of your body (towards your other hand or towards your thumb), finger position is low (down in the graph).

(3) If the finger is relaxed, finger position is 0.

(4) This graphing convention holds for both hands.

iii) Graph 1

(1) Assume the finger travels from resting, left, resting, right, resting in about 1 second.

(2) Graph this motion.

(3) You are encouraged to hand-draw all graphs.  The graphs may be approximate.

iv) Graph 2

(1) Assume the finger travels from resting, left, resting, right, resting in about ½ second.

(2) Assume that the amplitude of the motion decreases by ¼ as a result of this speed (frequency) increase.

(3) Graph this motion.

2)   Graphing two-finger motion.

a)    The paradigm.

i)     Hold both hands out in front of you with only your index fingers extended.

ii)   Put the back of your hands to the ceiling.

iii) Move your fingers left and right.

b)   Graph this finger movement.

i)     Fingers alternating.

(1) Move your fingers so that they are both to the right at the same time and both to the left at the same time.  For example, the right hand index finger will be to the outside at the same time the left hand index finger is to the center.

(2) Graph 3a

(a) Assume each finger travels from resting, left, resting, right, resting (or the reverse direction) in 1 second.

(b) Draw a graph as in Graph 1 for the motion of the right finger in black ink.

(c) On the same graph, graph the motion of the left finger in blue or red ink.

(d) Draw 7 seconds of motion.

(e) (Save space on the page, as you will be doubling the length of this graph. See the next graph.)

ii)   Fingers together.

(1) Move your fingers so that they both move to the center at the same time and both move to the outside at the same time.

(2) Graph 3b

(a) Repeat Graph 3a for this new alternating motion.

(b) Do this by continuing Graph 3a for an additional 7 seconds.

(c) Make the link between Graphs 3a and 3b as smooth as you can.

iii) Call Graph 3a and 3b together Graph 3.

c)    Relative phase.

i)     Graph 4

(1) Axes

(a) Time on the x-axis, 0-14 secs.

(b) Relative phase on the y-axis, 0 deg. to 360 deg. (or 0 to 2 pi).

(2) Look at Graph 3.

(a) If the peaks of the two curves (the two fingers) are in the same place at the same time the relative phase is 0 deg. 

(b) If the peaks of the two curves are in exactly opposite places at the same time, the relative phase is 180 deg (pi).

(c) Intermediate values can be obtained if the peaks are offset by different amounts (you may only see this at the intersection of Graphs 3a and 3b).

(d) On this new set of axes, plot the relative phase of Graph 3.

(e) (Save enough space under this graph to draw the next figures.)

d)   Potential wells and stable motions.

i)     Look at Fig. 5 from Haken, et al.

ii)   Draw 3 potential functions.

(1) Draw a potential with a stable state at 180 deg. out of phase.

(2) Draw a potential where a stable state at 180 deg. out of phase just disappeared.

(3) Draw a potential with a stable state only at 0 deg. out of phase.

iii) Draw arrows from each of these 3 figures to their corresponding locations in Graph 4.