We can get the first lyapunov coefficient of Hopf bifurcation by using MatCont.There I have a quite general question:
When I get the information about Hopf bifurcation which are printed in the MatLab command window：
label=H,x=(0.015189 -5.254172 -0.001297 0.074389 126.942956)
First Lyapunov coefficient=-9.231096e+01
Now,I want to get the coefficients of normal form in the vicinity of the Hopf bifurcation which are a,aa,c,cc as the following shows
A'=a(b-126.942956)A+cA^3
B'=aa(b-126.942956)+cc*A^2
where A is a measure of the amplitude of the born limit cycle
Is it possible in MatCont to get them(a,aa,c,cc),or other ways?
Thank you in advance,
Last edit: Bo 2016-12-09
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I am not entirely sure I understand your form (A',B') and whether it is correct. You cannot get them immediately. I can see that with some effort one could get such coefficients from adapting code in init_HH_NS1.m as this script involves the required multilinear forms. But this requires some dedicated programming and center manifold reduction techniques.
Best, Hil
From: Bo Zhu zbscorpion@users.sf.net
Sent: Friday, December 9, 2016 10:06 AM
To: [matcont:discussion]
Subject: [matcont:discussion] the normal form of Hopf bifurcation
We konw the first lyapunov coefficient of Hopf bifurcation by using MatCont.There I have a quite general questions:
When I get the information about Hopf bifurcation which are printed in the MatLab command window：
label=H,x=(0.015189 -5.254172 -0.001297 0.074389 126.942956)
First Lyapunov coefficient=-9.231096e+01
Now,I want to get the coefficients of normal form in the vicinity of the Hopf bifurcation which are a,aa,c,cc as the following shows
A'=a(b-126.942956)A+cA^3
B'=aa(b-126.942956)+cc*A^3
Is it possible in MatCont to get them(a,aa,c,cc),or other ways?
Thank you in advance
A is r,B is \theta
A'=a(b-126.942956)A+cA^3
B'=aa(b-126.942956)+cc*A^2
The above formulas which reduce four-dimensional system are getted by using the method of multiple scales
If you would like to refer to this comment somewhere else in this project, copy and paste the following link:
We can get the first lyapunov coefficient of Hopf bifurcation by using MatCont.There I have a quite general question:
When I get the information about Hopf bifurcation which are printed in the MatLab command window：
label=H,x=(0.015189 -5.254172 -0.001297 0.074389 126.942956)
First Lyapunov coefficient=-9.231096e+01
Now,I want to get the coefficients of normal form in the vicinity of the Hopf bifurcation which are a,aa,c,cc as the following shows
A'=a(b-126.942956)A+cA^3
B'=aa(b-126.942956)+cc*A^2
where A is a measure of the amplitude of the born limit cycle
Is it possible in MatCont to get them(a,aa,c,cc),or other ways?
Thank you in advance,
Last edit: Bo 2016-12-09
Dear Bo,
I am not entirely sure I understand your form (A',B') and whether it is correct. You cannot get them immediately. I can see that with some effort one could get such coefficients from adapting code in init_HH_NS1.m as this script involves the required multilinear forms. But this requires some dedicated programming and center manifold reduction techniques.
Best, Hil
From: Bo Zhu zbscorpion@users.sf.net
Sent: Friday, December 9, 2016 10:06 AM
To: [matcont:discussion]
Subject: [matcont:discussion] the normal form of Hopf bifurcation
We konw the first lyapunov coefficient of Hopf bifurcation by using MatCont.There I have a quite general questions:
When I get the information about Hopf bifurcation which are printed in the MatLab command window：
label=H,x=(0.015189 -5.254172 -0.001297 0.074389 126.942956)
First Lyapunov coefficient=-9.231096e+01
Now,I want to get the coefficients of normal form in the vicinity of the Hopf bifurcation which are a,aa,c,cc as the following shows
A'=a(b-126.942956)A+cA^3
B'=aa(b-126.942956)+cc*A^3
Is it possible in MatCont to get them(a,aa,c,cc),or other ways?
Thank you in advance
the normal form of Hopf bifurcation
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A is r,B is \theta
A'=a(b-126.942956)A+cA^3
B'=aa(b-126.942956)+cc*A^2
The above formulas which reduce four-dimensional system are getted by using the method of multiple scales
Last edit: Bo 2016-12-09