Continuation of homoclinic to saddle from a BT point
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Hello,
I have downloaded and installed MatCont7p6; I am using Matlab 2024b.
My system is:
X'=Lambd-betZ/(b+Z)XY-muuX
Y'=betZ/(b+Z)XY-alphZY-muuY
Z'=p(1-etalphZY/(a+alphZY))Z-(m_0+m_1Z)Z
where Lambd,bet,muu,alph,p,et,a,m_0,m_1 and b are parameters and X,Y,Z are coordinates, and I choose the first and second order derivatives to be calculated symbolically, and others are calculated numerically.
After the continuation of the limit point curve, I found two BT points(which I believe indicates the existence of a cusp type Bogdanov-Takens bifurcation since the numeric simulations have shown the existence of fold bifurcation of cycles) and the two parameter bifurcation diagram is attached.
The problem is I fail to plot the homoclinic bifurcation curve from any of the BT points.
After I choose the curve type ”Homoclinic to saddle(init_BT_Hom)”, the computation along each direction will lead to an Error in file “_multilinearforms.m” called “at least one other function definition was not closed”(the error is shown in the second figure).
I tried to add an “end” at the last of the file “*_multilinearforms.m” to let all functions are closed, and compute again.
This time, if I select the upper BT point with larger “bet” and “a”, I get a different error ”Dimensions of arrays being concatenated are not consistent”(shown in the 3rd fig), and the command window of MATLAB give a waring “Center manifold coefficients do not satisfy equations”( the 4th fig); if I select the lower BT point, the error disappear, but I still fail to do the continuation since the command window shows the warning and “current step size too small” (the 5th fig), and I can’t solve that by adjusting all three stepsizes.
The parameter values are in the last fig.
What should I do to get the homoclinic curve?
I sincerely hope you can provide guides and help. Thank you in advance for your time and consideration.
With best regards.
Finally some time to look into this. I can see the warnings&errors appear, but digging through I observe another problem. Your system is badly scaled, with such large values of the parameters and coordinates, and when I follow the code, I observe that the dimension of the stable manifold is zero. That leads to missing equations and hence the mismatch in dimensions.
Try to rescale your system so that the numerics do not suffer from that, or locate the BT-point with more precision.