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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 {Ωi }5i=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 {θi0 }5i=1 randomly from the uniform distribution on [−π/2, π/2]. Calculate Ke and choose a constant K > Ke . Run the codes for t from 0 to 10 and plot the curves for {θi (t)}5i=1 in the same frame. Use the specific numbers of your run to explain why all the assumptions in Theorems 2.10 and 2.11 are satisfied. 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 {Ωi }5i=1 and {θi0 }5i=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 Ωi = −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 satisfied. In (2.47) the first assumption is a restriction on the initial phase configurations. 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 defined 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 sufficiently 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 {θi0 }5i=1 are randomly chosen from the uniform distribution on [−π, π]. Keep {Ωi }5i=1 as randomly chosen from the uniform distribution on [−1, 1]. Keep {Ωi }5i=1 sorted as well. Set K = 0.2 and run the codes for t from 0 to 30. What are your D(θ0 ) and D(Ω)? 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 π < D(θ0 ) < 2π to be relevant to our discussion. Otherwise, rerun your codes. What is the value for K now? What are the values of D(θ0 ) and D(Ω)? 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.

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