Limit cycle stability

[Viktor Thunell]

Hi All,

I have three general question on analysis of limit cycles in MatCont.

• When assessing the stability of a limit cycle I understand that the first Lyapunov coefficent (FLC) can be used; a negative FLC corresponds to a supercritical or stable limit cycle and a positve FLC av subcritical or unstable limit cycle. Is this enough to be certain of stability or do I need further "proof" (like the values of the multipliers)?

• If I continue an equilbrium curve after reaching a Hopf bif. point, will MatCont detect changes in the stability of the limit cycle after this point? I.e. does the stability given by the FLC in the command window for the Hopf hold for the whole limit cycle parameter space?

• Furthermore, is the FLCstored in the .mat-file, in that case where?

Kind regards

Viktor

[hilmeijer]

1. For stability you need that the other eigenvalues all have negative real part.
2. No, the Hopf bifurcation provides you with local information (close to the bifurcation value). As the continuation tracks the equilibrium, it will not inform you about the limit cycle. You need to continue the limit cycle itself for that.
3. It is in the s-struct where all special info is stored.

Best, Hil

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From: Viktor Thunell vthunell@users.sourceforge.net
Sent: Tuesday, September 25, 2018 11:23 AM
To: [matcont:discussion]
Subject: [matcont:discussion] Limit cycle stability

Hi All,

I have three general question on analysis of limit cycles in MatCont.

• When assessing the stability of a limit cycle I understand that the first Lyapunov coefficent (FLP) can be used; a negative FLP corresponds to a supercritical or stable limit cycle and a positve FLP av subcritical or unstable limit cycle. Is this enough to be certain of stability or do I need further "proof" (like the values of the multipliers)?

• If I continue an equilbrium curve after reaching a Hopf bif. point, will MatCont detect changes in the stability of the limit cycle after this point? I.e. does the stability given by the FLP in the command window for the Hopf hold for the whole limit cycle parameter space?

• Furthermore, is the FLP stored in the .mat-file, in that case where?

Kind regards

Viktor

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Limit cycle stability

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[yanders]

Dear Hil
I want to plot the stable regions of my bifurcation diagram using solid lines and the unstable regions using dash lines. I drew Figure "plot 1", how does Matcont draw unstable regions with dashed lines, as shown in "plot 2"?
Best regards
yanders

[hilmeijer]

In MatCont7px, it is possible to plot the unstable branch of a LC or EP-curve with a different line style.
To do this, select in the Main window; "Options-->Plot Properties". Under
LC if unstable enter
'Color', 'blue', 'linestyle', ':' (replace the : by -. or -- if you like)

If the option is not there, then you have started MatCont with a existing system having an older session without this "LC if unstable" option. To get around that, do the following: First a clean start with

matcont 1 1 (just two random arguments)
Next enter a new system and then the option is available.

Best regards,
Hil

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From: yanders lyfyanger@users.sourceforge.net
Sent: Wednesday, July 8, 2020 11:18 AM
To: [matcont:discussion]
Subject: [matcont:discussion] Re: Limit cycle stability

Dear Hil
I want to plot the stable regions of my bifurcation diagram using solid lines and the unstable regions using dash lines. I drew Figure "plot 1", how does Matcont draw unstable regions with dashed lines, as shown in "plot 2"?
Best regards
yanders

Attachments:

• plot1.jpg (86.1 kB; image/jpeg)
• plot2.png (54.7 kB; image/png)

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Limit cycle stability

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[yanders]

Dear Hil
Thank you for your help.
Best regards
yanders

[Viktor Thunell]