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#60 Detecting many apparently non-existent BP and FP

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open
nobody
None
2024-05-08
2024-04-25
NK Liu
No

Hello!
I recently started trying to use the COCO toolbox for bifurcation analysis of power electronic converters (the periodic ODE of the converter can be found in the attached ODE_MMC_SIMP.m, which includes a single parameter Kp_CC). However, when performing the bifurcation analysis (corresponding code can be found in coco_bifurcation.m), the program detects many apparently non-existent BP and FP within a very small range near the initial Kp_CC. Can you please explain why this problem occurs and how to solve it? Thank you.

Best, NK

2 Attachments
ODE_MMC_SIMP.m
coco_bifurcation.m

Discussion

  • Harry Dankowicz

    Harry Dankowicz - 2024-04-25

    Hello NK.

    My recommendation is that you try to rebalance your variables so that they are of similar magnitude and vary on comparable scales.

    When I graph the solutions that COCO finds, the variables are of very different orders of magnitude and vary over very different orders of magnitude. This is not good practice when doing numerics.

    Since COCO uses various tolerances for identifying convergence and estimating errors, significant imbalance between variables renders the results insensitive to some variations and overly sensitive to others. The predictions should then be taken with a significant grain of salt.

    I hope that helps.

    /Harry

     

    • NK Liu

      NK Liu - 2024-04-29

      Hello again Harry!

      I have normalized the ODE as you suggested and made coordinate transformations for certain variables (the resulting ODE is the best I can think of at the moment to meet your requirements, it can be found in attached new ODE_MMC_SIMP.m ).

      I performed bifurcation analysis with three different ranges of Kp_CC (see attached coco_bifurcation.m):

      TEST1: Kp_CC belongs to [0.05, 0.6]. In this case, the program detected a torus bifurcation at Kp_CC=0.1129. By observing the corresponding eigenvalues and directly integrating the ODE with some selected parameters, I confirmed that the program's results are correct.

      TEST2: Kp_CC belongs to [0.6, 0.9]. The program didn't detect any bifurcations, but from the output eigenvalues, it can be observed that there are eigenvalues outside the unit circle at Kp_CC=0.9, indicating that there might be undetected torus bifurcations within [0.6, 0.9].

      TEST3: Kp_CC belongs to [0.05, 10]. In this case, the program only detected the torus bifurcation at Kp_CC=0.1129, and all connections between the EPs in the bifurcation diagram turned into dashed lines (which, according to my understanding, indicates that the periodic solution is unstable).

      I also tried using the Jacobians provided by symcoco (calculated using SYM_ODE.m), but the results were the same. How can I solve this issue?

       
      ODE_MMC_SIMP.m
      SYM_ODE.m
      coco_bifurcation.m

  • Harry Dankowicz

    Harry Dankowicz - 2024-05-08

    Hello NK,

    Sorry for the delayed response. I notice a couple of things when running your code:

    1. The periodic orbit found using ode15s includes a couple of variables that are of order 10^-7 and 10^-5 while all the other variables are of order 1. I still think that could cause numerical difficulties as the numerical routines use relative tolerances.
    2. The third component of your vector field equals 0. That seems to suggest that the periodic orbit that you compute is not isolated (since any value of y3 would work. I am not sure why you include this component if it is meant to be constant.

    I believe the effect of 2. is to render the continuation problem singular. Reliable continuation requires a regular continuation problem, one in which there is a unique periodic orbit for every local combination of parameter values.

    Best,

    Harry

     

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