How to create the window for displaying the change of eigenvalues every time BF runs

When calculating a bifurcation set on an arbitrary parameter plane using BF, a figure that shows the change of the positions of eigenvalues in the Gauss plane, i.e., root loci, is created in addition to the figure of a 2-parameter plane to show calculation results in real time. The figure is to monitor whether bifurcation points with co-dimension 2 occur during calculation of the bifurcation set. In the default settings, the previous results are cleared every time BF runs, and the root loci are shown in the same window.

You may want to see differences in the root loci under different parameter settings. In this case, change the configuration such that the display window starts every time BF runs.

1. Select Setting -> Graphic display.

2. Start up the configuration window.

3. Uncheck the checkbox single figure.

How to change the convergent precision of the Newton's method

When bifurcation sets are calcurated, information regarding precise coordinates of equilibrium points or periodic points of the Poincare map is necessary. Newton’s method is used to obtain these coordinates. The convergence precision is directly correlated to the accuracy of the calculations to obtain the bifurcation set.

During calculations, if the convergence precision of Newton’s method is inaccurate, it may give rise to problems such as the following:

• A parameter set that is not the true bifurcation set may be tracked.
• it may be failed the calculation of the value of a parameter when a bifurcation occurs.
• A parameter set that is not the true bifurcation set may be tracked.

In the default settings, when BF is initially started, the convergence precision is set to 1E-6. If a discontinuous point is included the set when a bifurcation set is tracking, users must be .... The behavior of BF may improve by increasing the convergence precision.

The following steps are used to change the convergence precision of Newton’s method.

1. Select Setting -> Newton's method.

2. Start up the configuration panel of Option.

3. Set the eps(and/or feps) to an appropriately precision.

How to change the divergent condition of the Newton's method

Currently, the following are used to determine convergence or divergence as the divergence criterion of Newton’s method*1:

1. Number of iterations (iter)
2. The sum of the elements in the Jacobian matrix used in Newton’s method (gmax)
3. The values of the state variables in the variational equation upon numerical integration (emax).

In particular, if the Newton’s method does not converge in the number of iterations iter, Newton’s method considers the system divergent or no solution. Therefore, iter must be appropriately changed depending on the problem. gmax and emax are used to terminate the program when abnormal values are detected internally.

In the default settings, when BF is initially started, values that are considered appropriate in general are configured:

• iter = 16
• gmax = 1E+10
• emax = 100 However, these values may have to be changed depending on the problem.

The following steps are used to change the divergence criteria for Newton’s method.

1. Select Setting -> Newton's method.

2. Start up the configuration panel of Option.

3. Change gmax (or emax) in the parameter for Newton's method to an appropriate precision.*2 Or, change iter, the number of iterations, to an appropriate number.

• In general the convergence of the Newton's method is quadratic, and therefore only a few number of iterations are necessary for convergence.
• If the number of the iter is too large, note that the solution may not be converged to the desired solution.

How to control the automatic switching of parameters

In calculating a bifurcation set on an arbitrary two-parameter plane, one unknown parameter and one control parameter are used. The bifurcation parameter value is calculated when the control parameter is fixed. Next, similar calculations are conducted when the control parameter is slightly changed. The procedure is repeated to obtain a set of parameters that result in bifurcation in the designated parameter plane.

Here, if the curvature of the parameter set is very large, the bifurcation set may be efficiently obtained by switching the configured unknown and control parameters. In the actual calculations, the curvature is calculated every time a bifurcation point is obtained.

auto sw gives the maximum curvature. If the curvature of the bifurcation set exceeds this value, the unknown and control parameters are swiched and the calculation of the bifurcation set is continued.

The following steps are used to change the criteria for parameter switching.

1. Select Setting -> Step size control.

2. Start up the configuration window.

3. Change auto sw in adaptive step size control to appropriate value.

• However, note that if a too small number is selected (＜１), the switching of control parameter happens frequently and may cause problems in tracking the bifurcation set.

How to stop the calculation of the inverse set

The bifurcation set is tracked from an initial bifurcation point. However, unless the set is ring-like, the calculation of the bifurcation set fails at some parameter. There are many possible causes, including:

• The bifurcation set that was tracked does not exist any more, since the set connected to another bifurcation set (i.e., the generation of the bifurcation point with co-dimension 2).
• Newton’s method did not converge because the point is singular, such as a cusp point.
• Trivial, but the parameter value reached the stop limit parameter value.

Here, after the calculations of a bifurcation set stopped, the next step is to calculate the bifurcation set from the initial bifurcation point with the parameters changing in the other direction. In the default settings of BF, after the calculation of the bifurcation set stopped for some reason, the bifurcation set is automatically tracked from the initial bifurcation point with the parameters changing in the other direction.

In many cases, the user only needs to obtain the bifurcation set in a given range, and therefore in some cases, calculation of the bifurcation set in the other direction is not necessary. Therefore, to stop the automatic tracking of the bifurcation set in the other direction, only the following switch needs to be turned off.

1. Select Setting -> Step size control.

2. Start up the configuration window.

3. Set the reverse calculation switch to Off.

How to change the numerical integration method(Caution !)

It is necessary to numerically integrate the solution when searching for bifurcation points of limit cycles and periodic solutions. Therefore, selecting what solver to use for numerical integration is an important issue. There may be some situations such that a stiff solver must be selected.

Basically, to find stable limit cycles or periodic solutions, simulation using PP is conducted to obtain attractor information. Information on the solver used at this stage is passed directly to FIX, therefore the numerical calculation is safer when the solver is not changed at this stage as much as possible.

However, if the convergence of the Newton’s method is bad, or does not converge, the match between the solver used and the problem may not be good. Then, it may need to try searching bifurcation points after changing the solver.

The following steps are used to change the solver.

1. Select Setting -> ODE solver.

2. Start up the configuration panel.

3. Select the solver in the parameter for ODE solver. For details of each solver, see for example, parameters for ODE solver.

• Caution: Even if the calculations successfully run by changing the solver, the numerical calculations are safer when PP is used to obtain attractor information under the same conditions, and then run FIX and BF.

How to improve the precision of numerical integration(Caution !)

As seen in How to change the numerical integration method, it is necessary to numerically integrate the solution when searching for bifurcation points of limit cycles and periodic solutions. The precision of the numerical integration is an important issue.

Basically, to find for stable limit cycles or periodic solutions, simulation using PP is conducted to obtain attractor information. Information on the solver used at this stage (including precision) is passed directly to FIX, and therefore the numerical calculation is safer when the precision is not changed at this stage as much as possible.

However, if the convergence of Newton’s method is bad, or does not converge, the configured precision may be too low. Then it may need to try searching bifurcation points after changing the precision.

The following steps are used to change the precision.

1. Select Setting -> ODE solver.

2. Start up the configuration panel.

3. Change the parameter Relative tolerance or Absolute tolerance for ODE solver. For details of each parameter, see for example, parameters for ODE solver.

• Caution: Even if the calculations successfully run by changing the solver, the numerical calculations are safer when PP is used to obtain attractor information under the same conditions, and then run FIX and BF.
• Caution 2: Increasing the precision of the numerical integration means the step size is decreased to keep the error from the true solution into the designated precision. Therefore, calculation speed must be sacrificed.

*1 For the meaning of iter, gmax, and emax, see for example, the BF manual: Newton's methods.
*2 In almost all cases, it may not need to change these values.

Last-modified: 2009-07-23 (Thu) 20:13:30 (3767d)