Comments to common questions and mistakes

    Important notice: users of DDE-BIFTOOL are assumed to have a basic knowledge on bifurcation analysis of dynamical systems and on the nature of delay equations. The manual on the package does not replace the corresponding text books! To avoid trivial questions and mistakes, users must read the manual carefully and run the demo example(s) before analyzing their own DDE systems.

  • Results obtained with DDE-BIFTOOL "contradict" to the ones obtained with a time integration software (e.g., dde23, XPPAUT)

    • The accuracy of results depends on the accuracy of the used numerical method and/or the software package. Usually, the accuracy is determined by a default value of the minimal step size of the discretization or by default values of certain tolerances. These default values are available to users. To be sure that the obtained result is correct, one has to decrease the default values and redo the computations.
    • It is practically impossible to obtain bifurcation points (i.e. bifurcation values of the model parameters) accurately using time integration. In a neighborhood of a bifurcation point, jumps of the computed trajectory to a "more" stable solution usually occur. Furthermore, the result of time integration depends on the initial data, the accuracy used and on the time of integration as well.
    • With respect to stability of stationary solutions, DDE-BIFTOOL gives two sets of the rightmost (dominant) characteristic roots: (1) the computed roots and (2) their corrections (by a Newton iteration on the associated characteristic matrix). If you are in doubt, you can check whether the roots satisfy the characteristic equation for your DDE system. Be sure that the analytical derivatives, which you use in the characteristic equation, are CORRECT.

  • "TIME_H warning: h_min is reached"

    • Such warnings indicate that the time integration step required to obtain good approximations to the requested rightmost characteristic roots is too small. By default, characteristic roots are computed up to the roots with real part greater than -1/(maximal delay), cf. p. 23,24 of the manual. DDE-BIFTOOL gives two sets of the rightmost characteristic roots: (1) the computed roots and (2) their corrections (by a Newton iteration on the associated characteristic matrix). If, in your case, the corrected dominant roots are close to the corresponding computed roots (as in the first demo example, see the manual), just accept this warning "as is", the corrected roots are correct, no doubt.
    • If the most dominant (the first few) roots were computed with a very bad accuracy, i.e., their corrections were not computed at all, or the corrected dominant roots are not close to the computed roots, you can try the following. First, be sure that all analytical derivatives, which you defined in sys_deri.m, are correct and that your system is properly scaled (see below). Then, you can decrease the value of the minimal time integration step, say h, to obtain a better approximation to the rightmost characteristic roots, as follows. The default value of h is 0.01. Decrease h, e.g., by setting

      h=0.005;

      and define

      method.stability.minimal_time_step=h;

      in your running code, after the line

      method=df_mthod('stst');

      Note that the smaller h, the larger the size of the required memory and the computational time.

      If this procedure does not help and you are sure that there are no mistakes in your files, mail all your system definition files (all "sys" files) and the code you run, so that we can reproduce your results and try to help you.

  • System is not properly scaled

      Be sure that the system parameters and unknowns are properly scaled, i.e. the ratio of their maximal and minimal values is not too large. Otherwise, you can encounter difficulties while computing stability of stationary and periodic solutions and branches of these solutions via continuation. For example, avoid a situation where the values of the system unknowns are about 10000 and the free parameter (you compute a branch of solutions w.r.t. this parameter) varies in the interval [0,1].

  • Problems due to differentiation of algebraic or integral equations

      When a model is described by a system of differential equations coupled with algebraic or integral equations, differentiation of the latter is used, by some researchers, to analyze the obtained DDE system with DDE-BIFTOOL. However, due to the differentiation, an isolated solution of the original system is not an isolated solution of the resulting DDE system. The algebraic (or integral) equations form an invariant set for the latter and they must be taken into account when analyzing the obtained DDE system. For this, one has to program the corresponding (to the invariant set) algebraic conditions in the code sys_cond.m which is specifically designed for algebraic conditions. In case of stationary solutions, this can be done analogously to the cases described in p. 52 of the manual. In case of periodic solutions, one has to program, in sys_cond.m, a collocation equation. This is a nontrivial task and it is not recommended to users who are not familiar with routines of the package. Note also that, due to the differentiation, the characteristic equation and the monodromy operator for the DDE have extra zero roots, respectively multipliers at 1, the number of which equals to the number of the algebraic equations.

  • How to compute and plot stability along a branch of periodic solutions?

      Analogously to the case of a branch of steady state solutions, explained in p.21,22 of the manual, namely,

      branch4=br_stabl(branch4,0,1);
      [xm,ym]=df_measr(1,branch4);
      figure(1);clf;
      br_plot(branch4,xm,ym,'b');

      Here branch4 is the computed branch of periodic solutions. Note that computing stability of all periodic solutions along a branch takes considerably more computational time than in the case of stationary solutions. Look at the meaning of the 2-nd and 3-rd input parameters of the code br_stabl.m. You can decrease the cost by computing stability of some solutions only.

  • The first point along a branch

      Note that the first point (only the first point!) along a branch (branch1 here) is defined as

      branch1.point=stst;

      The definition

      branch1.point(1)=stst;

      works in older versions of Matlab and it does not work in the latest versions.

  • Comments to certain parts of the demo examples

    • The line

      branch1.parameter.min_bound(4,:)=[4 0];

      in the first demo example has the following meaning. Let the system under study has three delays (as in the first demo example). As it is implemented in the package, the first (here three) "minimal bounds" correspond to the delays (they are set to 0 by default). The next "minimal bound", here the 4-th one, is specified by the user for the free parameter. The second "4" in the line indicates the number of the free parameter. That is, these two "4" have different meaning.

    Some recommendations


  • When you compute (correct) steady state and periodic solutions, you can use

    format long e;
    method.point.print_residual_info=1;
    method.point.newton_max_iterations=10;
    method.point.newton_nmon_iterations=3;

    to be sure that the corrections converge, i.e. that your computations are correct. If there is no convergence (the residual values are increasing), check your codes for a mistake.

  • If "something" goes wrong, check whether all analytical derivatives, which you defined in sys_deri.m code, are correct. For this, you can compare the "output" of your sys_deri.m with the corresponding "output" of the default routine df_deriv.m, where the derivatives are computed numerically, by finite difference formulas, cf. p. 6,7 of the manual. Note that, to avoid typing mistakes, it is not very important to define second order analytical derivatives, you can use numerical derivatives instead.

  • In case you met a problem and (a) you checked all your system definition files ("sys" files) for mistakes, (b) the above comments have not helped you to solve the problem, then mail us all your system definition files and the code you run, so that we can reproduce your results and try to help you.