Good morning, MatCont team, and thank you for the amazing work you have done. I'm pretty new to numerical continuation techniques, but I think my computation is mostly correct. I just need help with the last part of the bifurcation analysis. I have implemented a large nonlinear system of 50+ equations in the following form (i = 1...50):
Qa_i' = Qb_i
Qb_i' = f(Qa_i,Qb_i,L,coefficients)
I can't report the entire system since it is close to 120 pages long. The coefficients magnitude range is from very small (1e-15) to very large (1e+8), and my only free parameter is L. The process that I followed was:
Found the Hopf point with an equilibrium analysis (at L = 520) (exactly where it should be!)
Started limit cycle continuation from the Hopf point up to L = 590, where the code detected a LPC (where it should be!)
Selected this LPC and started a continuation of the limit cycle. The next LPC was found at L = 490 (as I expected it!)
Here's my problem! I want to find the unstable branch from L = 490 to the Hopf point at L = 520 (I have attached figures for reference). If I start the limit cycle analysis from L = 490, the system goes back to L = 590. Am I doing something wrong? How can I proceed?
Step 4: perhaps clear for you what you want, but not for me. You want to recompute the whole branch from the 2nd LPC to the first LPC to the Hopf, or you want to follow the branch beyond the turn? Why do steps 2&3 (stopping and restarting) at all? You could extend the curve.
If I look at the amplitude db6.jpg, it is worth considering smaller stepsizes to get something more smooth. BTW for me it's midnight.
Good luck. Hil
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Thank you for the reply! Yes, I will use smaller step sizes in the final simulation. My goal for this one is to find the LPC points and their branches.
- Extending the curve works too! It was midnight for me as well :)
- I need to follow the branch beyond the turn at LPC = 490 down to H = 520. However, extending the simulation doesn't work.
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>>However, extending the simulation doesn't work.
It is not a simulation (timestepping), this is about continuation. The question is why extending does not work. Describe what happens? Also with the whole run starting from the Hopf, without restarting, but just extending (or set MaxNumPoints to 5000).
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The simulation from the Hopf point (L=520) goes to LPC_1 (L=590), then proceeds to LPC_2 (L=490). If I further extend the simulation, the LC increases to L > 1200. I did not complete the simulation, so I don't know if it reaches another LPC, but from another numerical method that I tried, I should find a branch from LPC_2 to the Hopf point.
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Just repeating to fix terminology, I think you're doing continuation, not timestepping when computing that branch.
Suppose some part of the branch with that other method corresponds to stable oscillations. Then you could use that as a starting point for numerical continuation of possibly another branch. I do not know your model but there is no guarantee that all solution branches are connected the way you expect.
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Last question.. Looking at the blue lines in the attached plot, with the x-axis representing my parameter and the y-axis representing the coordinate, I found that the branch between LPC1 and LPC2 is unstable (Multiplier > 1). Is it possible that, starting from LPC2, I can't find an additional unstable branch directed towards the Hopf point because the stable branch that starts at LPC2 'attracts' the continuation?"
There's not such force to deroute the continuation. Your graph looks fully natural to me. What I would do instead: Very carefully inspect your equilibrium and see if you have two Hopf bifurcation points.
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Good morning, MatCont team, and thank you for the amazing work you have done. I'm pretty new to numerical continuation techniques, but I think my computation is mostly correct. I just need help with the last part of the bifurcation analysis. I have implemented a large nonlinear system of 50+ equations in the following form (i = 1...50):
Qa_i' = Qb_i
Qb_i' = f(Qa_i,Qb_i,L,coefficients)
I can't report the entire system since it is close to 120 pages long. The coefficients magnitude range is from very small (1e-15) to very large (1e+8), and my only free parameter is L. The process that I followed was:
Thank you so much for your time and assistance!
Last edit: Lorenzo de Dominicis 2024-06-11
Step 4: perhaps clear for you what you want, but not for me. You want to recompute the whole branch from the 2nd LPC to the first LPC to the Hopf, or you want to follow the branch beyond the turn? Why do steps 2&3 (stopping and restarting) at all? You could extend the curve.
If I look at the amplitude db6.jpg, it is worth considering smaller stepsizes to get something more smooth. BTW for me it's midnight.
Good luck. Hil
Thank you for the reply! Yes, I will use smaller step sizes in the final simulation. My goal for this one is to find the LPC points and their branches.
- Extending the curve works too! It was midnight for me as well :)
- I need to follow the branch beyond the turn at LPC = 490 down to H = 520. However, extending the simulation doesn't work.
>>However, extending the simulation doesn't work.
It is not a simulation (timestepping), this is about continuation. The question is why extending does not work. Describe what happens? Also with the whole run starting from the Hopf, without restarting, but just extending (or set MaxNumPoints to 5000).
The simulation from the Hopf point (L=520) goes to LPC_1 (L=590), then proceeds to LPC_2 (L=490). If I further extend the simulation, the LC increases to L > 1200. I did not complete the simulation, so I don't know if it reaches another LPC, but from another numerical method that I tried, I should find a branch from LPC_2 to the Hopf point.
Just repeating to fix terminology, I think you're doing continuation, not timestepping when computing that branch.
Suppose some part of the branch with that other method corresponds to stable oscillations. Then you could use that as a starting point for numerical continuation of possibly another branch. I do not know your model but there is no guarantee that all solution branches are connected the way you expect.
Last question.. Looking at the blue lines in the attached plot, with the x-axis representing my parameter and the y-axis representing the coordinate, I found that the branch between LPC1 and LPC2 is unstable (Multiplier > 1). Is it possible that, starting from LPC2, I can't find an additional unstable branch directed towards the Hopf point because the stable branch that starts at LPC2 'attracts' the continuation?"
There's not such force to deroute the continuation. Your graph looks fully natural to me. What I would do instead: Very carefully inspect your equilibrium and see if you have two Hopf bifurcation points.