I am very new to mathematica and also to Hopf bifurcation or any bifurcation for that matter. But I am trying to obtain a Hopf bifurcation for a dynamical system. Now, so far, I cannot find any Q&As related to Hopf Bifurcation on this site.

The system of equations (1-2) are shown below

enter image description here

for which i have coded it as

Module[{nds, c, p}, 
nds =
NDSolve[{c'[t] == (v1 (p[t]/(p[t] + d1))^3 (c[t]/(c[t] + f5))^3 (N_open/N_total)^3)(ctot - (c1 + 1) c[t]) - v3c[t]^2/(k3^2 + c[t]^2),
         p'[t] == v4 (c[t] + (1 - a_lpha) k4/c[t] + k4) - Irp[t]}, {c,p}, {t, 0,100}]]]

and the parameters are as follows

enter image description here

My goal is to obtain the Hopf bifurcation similar to this

enter image description here

with C and P on the x- and y-axis respectively.

Can anyone guide me on how to obtain the Hopf Bifurcation. (Even the text book by Stan Wagon only describes the general Bifurcation)


closed as off-topic by xyz, user9660, Yves Klett, MarcoB, rhermans Jan 11 '16 at 14:59

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  • 1
    $\begingroup$ If C and P were on the axes then it would be what is called a phase portrait of this dynamical system. A bifurcation diagram on the other hand would have either C or P on the y axis and a constant on the x axis, all other constants being fixed. The lines would correspond to equilibrium points, i.e. roots of the right hand side of the dynamical system. Please check your source again to find out what the axes represent and also please write out the equations in Mathematica syntax. $\endgroup$ – C. E. Jan 11 '16 at 6:36
  • $\begingroup$ @Pickett Thanks for the reply. I checked the source and the axes are as I have mentioned them. Solid line indicates stable steady states, dashed indicates unstable steady states, and dashed-dotted line gives the maximum and minimum. The system generates oscillations of C (variations in C's concentrations to be precise) with either a fixed P or a range of P. The bifurcation diagrams shows us the range of P that allows the oscillations to occur. Also like I said I am new to mathematica so it will take me some time to write down the ODEs in Mathematica syntax. $\endgroup$ – nashynash Jan 11 '16 at 6:49
  • $\begingroup$ You may do several things about it with Mma, but first of all you surely should learn, what is the Hopf bifurcation. There are lots of sources. $\endgroup$ – Alexei Boulbitch Jan 11 '16 at 8:27
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    $\begingroup$ If you wish to solve this problem with Mathematica, please write it in Mathematica format and define all quantities. As it stands, many symbols are undefined. Once that is done, use ParameticNDSolveValue with a suitably selected parameter. $\endgroup$ – bbgodfrey Jan 11 '16 at 16:43
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    $\begingroup$ @bbgodfrey So am I wrong in what I wrote? NDSolve will calculate trajectories, yes, but the Hopf bifurcation (and the bifurcation diagram) is about how the number of equilibrium points and their stabilities change over time. He shouldn't have to solve the system to see that, finding the roots and the eigenvalues of the system should be enough. (And yes, I know that the OP said that P and C are on the axes which would mean that it shows trajectories, but I don't believe that's right. This looks very much like a bifurcation diagram to me, with dotted lines for unstable equilibriums.) $\endgroup$ – C. E. Jan 11 '16 at 17:23