BF Options




Graphic display

  • Graphic display : It is possible to display in real time the computed results of the bifurcation set. The single figure option controls whether a new display window is created or not at the beginning of computations.

    index.php?plugin=attach&refer=BFatNonautonomous&openfile=figure.png

    • On : checked
      • Display the results on one window for all computations.
    • Off : unchecked
      • Create a new window every computation. To be used for results comparison, for instance.




Newton's method

  • parameters for Newton's method : Each bifurcation set are calculated to solve the simultaneous equation consisting of the equilibrium point equation or the fixed point one and the characteristic equation by using Newton's method. Each of the following items is a control parameter of this Newton method.


    index.php?plugin=attach&refer=BFatNonautonomous&openfile=newton.png


    gmaxIf the sum of all the absolute value of the derivatives with respect to the variables exceeds this value, the program considers it diverged.
    emaxIf the sum of all the absolute value of the system variables used in the Newton method exceeds this value, the program considers it diverged.
    epsIf the sum of all the absolute value of the differential values of the system variables used in the Newton method goes under this value, the program considers it converged.
    fepsIf the sum of all the absolute value of the system variables used in the Newton method goes under this value, the program considers it converged.
    iterNumber of convergences for the Newton method. One exceeded this number, the program halts.




ODE solver

  • parameters for ODE solver : Selection and detailed settings of numerical integration.
    • Solver:selection of ODE solver built in MATLAB is possible.

      index.php?plugin=attach&refer=BFatNonautonomous&openfile=solver.png


      ode45This is based on an explicit Runge-Kutta formula, the Dormand-Prince pair. It is a one-step solver; that is, in computing y(tn), it needs only the solution at the immediately preceding time point, y(tn-1). In general, ode45 is the best solver to apply as a "first try" for most problems.
      ode23This is also based on an explicit Runge-Kutta proposed by Bogacki and Shampine. It may be more efficient than ode45 at crude tolerances and in the presence of mild stiffness. ode23 is a one-step solver.
      ode113This is a variable order Adams-Bashforth-Moulton PECE solver. It may be more efficient than ode45 at stringent tolerances. ode113 is a multi-step solver; that is, it normally needs the solutions at several preceding time points to compute the current solution.
      ode23s};This is based on a modified Rosenbrock formula of order 2. Because it is a one-step solver, it may be more efficient than ode15s at crude tolerances. It can solve some kinds of stiff problems for which ode15s is not effective.
      ode15sThis is a variable order solver based on the numerical differentiation formulas (NDFs). These are related to but are more efficient than the backward differentiation formulas, BDFs (also known as Gear's method). Like ode113, ode15s is a multi-step method solver. If you suspect that a problem is stiff or if ode45 failed or was very inefficient, try ode15s.
      myselfThis is a standard fixed step 4th order Runge Kutta method.It is not recommended unless under particular circumstances.

      index.php?plugin=attach&refer=BFatNonautonomous&openfile=solver.png

    • Initial step : The initial step size that the ODE solver uses. When using one of the pre-installed solvers, since the step size is automatically adjusted, setting an initial step size unreasonably small (enter a value of the input field too big) will result in poor computation efficiency. When selecting the myself method, its step size is simply 1/(the value of the input field).
    • Relative tolerance : measures the error relative to the size of each state. The relative tolerance represents a percentage of the state's value.
    • Absolute tolerance : represents the acceptable error as the value of the measured state approaches zero*1.




Step size control


index.php?plugin=attach&refer=BFatNonautonomous&openfile=stepsize.png

  • reverse calculation switch : When starting the computation of the bifurcation set, the initial direction is considered to be along the parameter selected for the Y axis. If, for some reason, the computation stopped, this switch determines whether the following computation operates the opposite direction by using a negative step size at the initial point.

    index.php?plugin=attach&refer=BFatNonautonomous&openfile=reverse.png


    • On : When the computation stops, try to calculate the bifurcation set in the opposite direction
    • Off : When the computation stops, stop the whole BF program.

  • adaptive step size control : When tracing a bifurcation diagram, there are cases when adapting the parameters' variation step size to the bending of the curve can increase calculation efficiency. This switch enables or disables the step size control option.

    index.php?plugin=attach&refer=BFatNonautonomous&openfile=adaptsize.png



    • On : Use a variable step size
    • Off : Use a fixed step size(Default setting)*2
    • auto sw : While seeking for the value of the X parameter selected as the unknown variable by varying the parameter selected as Y, if the amount of the variation of the X parameter exceeds auto sw * step size, the computations are continued by switching the control for the X and the Y parameters.


      Symmetrical property

  • Symmetrical property:Definition of symmetrical properties of limit cycles and the detailed settings (for details see Ref.[1].)


index.php?plugin=attach&refer=BFatNonautonomous&openfile=symmetry.png

  • Not use this option: Out of consideration of a symmetrical property of limit cycles, i.e., normal action of BF.
  • Inversion:The case of limit cycles with an inversion symmetry. Thus, the limit cycle have the following property:

    index.php?plugin=attach&refer=BFatNonautonomous&openfile=eq_img01.gif.

    • mu = 1: This case corresponds to the pitch-fork bifurcation that the symmetrical property is preserved.
    • mu = -1: This case corresponds to the pitch-fork bifurcation that the symmetrical property is broken.
  • Permutation:The case of limit cycles with a permutation symmetry and a time shift. The limit cycle is an invariant under the operation by the following matrix and by the time shift:

    index.php?plugin=attach&refer=BFatNonautonomous&openfile=eq_img02.gif,

    where index.php?plugin=attach&refer=BFatNonautonomous&openfile=eq_img03.gif indicates the following matrix:

    index.php?plugin=attach&refer=BFatNonautonomous&openfile=eq_img04.gif.

    • mu=1:This case corresponds to the pitch-fork bifurcation that the symmetrical property is preserved.
    • mu = -1: This case corresponds to the pitch-fork bifurcation that the symmetrical property is broken.
  • Arbitrary:The case of limit cycle with an arbitrary symmetrical property. User must define the arbitrary symmetrical matrix.*3
    • divided number:When the symmetry operation is carried out to a limit cycle, you must also take into account the time shift([1]).This option is given as a divided number that need to shift how long time for the period of the limit cycle.

References


  • [1] Y. Katsuta and H. Kawakami,Bifurcations of periodic solutions in a nonlinear nonautonomous system with symmetry, Electronics and communications in Japan, Scripta Technica, Inc., Part 3, vol.77, no.3, pp.106--116, 1994.


Arbitrary eigenvalue

  • Arbitrary eigenvalue:BF tool can be traced not only some kinds of bifurcation sets, but also parameter sets with an arbitrary eigenvalue. To select this option, users are able to configure the value of the eigenvalue of the parameter sets to be traced.

    index.php?plugin=attach&refer=BFatNonautonomous&openfile=arbitrary.png




*1 for detail see odeset
*2 We strongly recommend to set off on this option at this time.
*3 Caution !: If this option is selected, users must describe the value of the eigenvalue to trace the pitch-fork bifurcation into the following Arbitrary eigenvalue option.

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Last-modified: 2009-07-23 (Thu) 20:13:30 (3374d)