For example, the nonlinear eigenvalue problem is $\lambda I -A_1-A_2e^{-\lambda}=0
$
For the scalar case, this can be done without software, see e.g. Section XI.2 of Delay Equations by Diekmann, Van Gils, Verduyn Lunel and Walther, 1995, as well as the references in Section XI.5. Also, see the book by Bellman and Cooke.
As far as the problem of interest arises as the characteristic matrix of a delay differential equation (this is the case for the example you give), a somewhat older MATLAB software is discussed in: D. Breda, S. Maset, and R. Vermiglio, TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations, 2009. The article and the software are freely available. Maybe Francesca Scarabel can tell you more about recent progress on this work.
Recently, also also the Julia package NEP-PACK became available, and it may be worth to have a look at it.
Thank you for your response. In NEP-PACK they have talked about DDE-BIFTOOL. I am unable to understand how to use DDE-BIFTOOL in Matlab. Some demos are there but I could not understand how to implement it in Matlab.
I see! Is there a specific difficulty that you experience with running DDE-BIFTOOL in MATLAB?
My colleague Maikel Bosschaert works a lot on the DDE-BIFTOOL code and has revised some tutorials and added others. I tag him here and expect that he will at least be able to point you to the most recent versions of the demos and the tutorials.
(You can also look at the tutorial supplement in our Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations, SIADS, 2020, but that is mostly about branch switching at codimension-two points and not so much about the nonlinear eigenvalue problem, which conceptually comes before bifurcation theory.)
Thanks again for your time, sir. I have looked into the supplementary materials in which a lot of Matlab implementations are given and which can be very helpful for the other researchers. Now If we have the following DDE of the form
$\dot{x}(t) = A_0 x(t) + A_1x(t-1)$, where $A_0 =
\left[ {\begin{array}{cc}
-5 & 1 \\
2 & 6 \\
\end{array} } \right]$ and $A_1
\left[ {\begin{array}{cc}
-2 & 1 \\
4 & -1 \\
\end{array} } \right]$ then the characteristic matrix is $\lambda I - A_0 -A_1 e^{-\lambda}$ , then which functions we need from DDE-BIFTOOL to compute the spectrum of this problem? If any of you Sebastiaan Janssens sir or Maikel Bosschaert sir have any suggestion regarding this problem I would be greatful to you. Thanks in advance.
It looks like what you may need is:
Tutorial II from https://sites.google.com/a/uhasselt.be/maikel-bosschaert/home
I would start in Section 5, assuming DDE-BIFTOOL has been set up correctly, otherwise review that from the sections before.
More specifically, I would beging by trying to reproduce the stability example from Section 5.3. The function you need (p_stabil) is called at the bottom of p.11 there, to compute a part of the spectrum of the steady state (stst) and save the obtained stability information in the stst-structure.
Note from there that the argument method.stability of p_stabil can be used to pass certain options to the spectrum calculation:
- Since the spectrum is not bounded in the left half plane, you will either go with the default value of minimal_real_part or have to set it by hand, as is done in Section 5.3.
- The semigroup theory of (classical) DDEs predicts that in every right half plane, the spectrum has only finitely many points. The parameter max_number_of_eigenvalues imposes an upper bound on how many of these are computed by p_stabil.
Does this get you going? Please try to work through the tutorial example first, because you can compare your output with the output in the tutorial. After this, I believe you will be able to also tackle your own example.
I hope the tutorials help you to understand how to use DDE-BifTool. Here is a short script to get you started for your particular system:
https://github.com/DDE-BifTool/DDE-Biftool/blob/master/demos/GyanSwarupNag/GyanSwarupNag.ipynb
As Sebastiaan Janssens mentioned, you may need to change certain options in the method_stst.stability struct.
Thanks a lot to both of you for your time. I have understood how to run the demo codes which are given in DDE-Biftool demos. I think that for the different problems we have to generate the @neuron_sys_deri differently. I do not understand this particular step from the script given in Github "Differentiate and generate code, exporting it to sym_GyanSwarupNag_mf (multi-linear forms)". Can you please elaborate?
Thanks in advance.
This particular step creates the file sym_GyanSwarupNag_mf.m, which contains the definition of the system and its (higher order) derivatives. This step is normally only done once, since it can be computationally expensive. In the system you are considering all higher order derivatives vanish, which can be verified by inspecting the file sym_GyanSwarupNag_mf.m. Therefore, in this case, it is not expensive.
Also, since you are only interested in the spectrum, the higher order derivatives are not used. These are needed for calculating normal form coefficients and certain continuation problems.
After you have created the system file with the function dde_sym2funcs you do not need to generate a file @neuron_sys_deri anymore. The field sys_deri of the funcs struct points to - via a wrapper function - the system file.
Let me know if anything is still unclear.
Thank you to both of you, for helping me to understand DDE-BIFTOOL. Now I am able to draw spectrum for different problems.
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