Differential Equations

Modeling with PDE, MA 461
Assignment 7: Kuramoto Oscillators
Due date: 03/15/2020
This assignment is to verify/explore properties of Kuramoto Oscillators governed by
(2.40) in the lecture notes. Hand in all plots with your answers/discussions.
1. Use the codes in the lecture notes to verify Theorems 2.10 and 2.11: Take 5 oscillators
with natural frequencies f
ig5i
=1 randomly chosen from the uniform distribution on
[?1; 1]. Sort the oscillators so you are able to keep track of them when verifying
Theorem 2.11. Also take initial phase values fi0g5i
=1 randomly from the uniform
distribution on [?=2; =2]. Calculate Ke and choose a constant K > Ke. Run the
codes for t from 0 to 10 and plot the curves for fi(t)g5i
=1 in the same frame.
Use the specic numbers of your run to explain why all the assumptions in
Theorems 2.10 and 2.11 are satised. Then use your plot to verify the conclusions in
the two theorems. You may not have the same plot as Figure 2.4 in the notes since
f
ig5i
=1 and fi0g5i
=1 have been randomly chosen but your plot should still agree with
the theorems.
2. In the class we discussed the concept of asymptotic complete phase synchronization
but concluded that it could only happen to identical oscillators. Modify your codes
such that all
i = ?0:5 but keep other parameters. In particular, the initial phase
values should be randomly chosen. Run the code. Are you able to see the phase
synchronization? If not, increase the upper bound of t from 10 to 20 or 30 until you
can see convergence. Discuss your result.
3. Next we explore what happens if the assumptions of Theorems 2.10 and 2.11 are not
satised. In (2.47) the rst assumption is a restriction on the initial phase congura-
tions. We remove it and allow generic initial phases. The second assumption in (2.47)
is for nonidentical oscillations which we keep. The last assumption in (2.47) becomes
irrelevant since Ke dened by (2.46) can be negative, thus we set K as a positive
constant instead. In such a situation Theorems 2.10 and 2.11 do not provide us any
information. However, in a new paper by S.-Y. Ha et al (to appear in Comm. Math.
Sci.) it is shown that if K > 0 is suciently large, asymptotic complete frequency
synchronization can be achieved. (There is no information on how large K needs to
be in that paper though.)
Modify your codes in Problem 1 such that fi0g5i
=1 are randomly chosen from the
uniform distribution on [?; ]. Keep f
ig5i
=1 as randomly chosen from the uniform
distribution on [?1; 1]. Keep f
ig5i
=1 sorted as well. Set K = 0:2 and run the codes for t
from 0 to 30. What are your D(0) and D(
)? Do you see frequency synchronization?
If you do not see frequency synchronization, increase K to 0.5, 1.0, … until your
plot shows clear synchronization. Make sure  < D(0) < 2 to be relevant to our
discussion. Otherwise, rerun your codes. What is the value for K now? What are the
values of D(0) and D(
)? Are the oscillators in the order according to their natural
frequencies (as described in the conclusion of Theorem 2.11)?
Give a discussion of your results.