How to change the action when detects a bifurcation point

The user can select the subsequent operation after a bifurcation point is detected. The following three patterns are available:

• Case 1: When a bifurcation point is detected, the information is displayed, but FIX does not terminate.
• Case 2: When a bifurcation point is detected, FIX terminates.
• Case 3: When a bifurcation point is detected, the step size of the parameters are controlled to obtain detailed parameter values when bifurcation occurs.

The procedure to terminate FIX can be configured using the following steps.

1. Select Setting -> Detection.

2. Start up the configuration window.

3. Select the terminated switch.

• I： Corresponds to Case 1. FIX does not terminate when a bifurcation point is detected.
• II： Corresponds to Case 2. FIX terminates when a bifurcation point is detected.
• III： Corresponds to Case 3. FIX continues calculation when a bifurcation point is detected. The step size is changed to obtain more accurate parameter values near the bifurcation point. FIX terminates after confirming that the parameter is close enough to the bifurcation condition.*1

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

When bifurcation points are searched using FIX, a figure that shows the change of the positions of eigenvalues in the Gauss plane, i.e., root loci, is created. In the default settings, the previous results are cleared every time FIX 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 FIX runs.

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

When bifurcation points are detected, 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 parameters that cause bifurcation.

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

• Initial value of the state variables and parameters when bifurcation occurs cannot be automatically obtained.
• A state that is not the true solution may be tracked.

In the default settings, when FIX is initially started, the convergence precision is set to 1E-6. If the terminated mode when bifurcations occured is set to Case 2 (see How to change the action when detects a bifurcation point), and in case FIX does not terminate when the bifurcation points are searched, the behavior of FIX 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 conditions of the Newton's method

Currently, the following are used to determine convergence or divergence as the divergence criterion of Newton’s method*2:

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 FIX 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.*3 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.
• When detecting saddle-node bifurcation, it may be better to choose a smaller number of the iter.

How to change the initial divided number

Quicker convergence can be achieved if previously obtained results are extrapolated and used as the initial value for Newton’s method. Approximation using a 5th-order polynominal equation is used to extrapolate. However, the extrapolation algorithm cannot be used until the first five points on the curve are obtained. As such, parameters must be partitioned between the first and second points. The divided number is defined as nnn.

In the default settings, the nnn is ten. Namely the step size between the first point of the parameter and the second one is divided into 10 partitions.

The following steps are used to change the divided number.

1. Select Setting -> Newton's method.

2. Start up the configuration window.

3. Change nnn in the parameter for Newton's method to an appropriate number. (If a number smaller than 5 is designated, 5 is assigned internally regardless of the initial designation.).

How to change the detection precision of bifurcation point

The parameter value where Newton’s method stops depends on the step size. If the parameter value is far from the true bifurcation point (in other words, the error between the eigenvalue when the bifurcation occurs and the eigenvalue when FIX stopped or terminated is large), tracking of the bifurcation curve using BF may fail.

If the terminated mode is set to Case III (see How to change the action when detects a bifurcation point.) during bifurcation search, then a bifurcation point is detected and Newton’s method diverges, the parameter step size is halved and tracking of the bifurcation point is resumed. After a few iterations, a good enough approximate value of the parameter which causes bifurcation can be obtained.

Seig_tol is used to determine whether to stop the program because a good enough approximate value is obtained. In other words, for an eigenvalue where bifurcation occurs, if the eigenvalue for the current parameters is μ, the program stops when

｜μ* - μ｜< Seig_tol

Seig_tol may have to be adjusted to obtain the initial bifurcation point, depending on the problems. The following steps are used to change this stopping criterion.

1. Select Setting -> Detection.

2. Start up the configuration window.

3. Change SeigTol_val to an appropriate number. (In the default settings, SeigTol_val is set to 0.01. Therefore the error from the true bifurcation point becomes within 1% are continued the calculation.)

• For example, for the saddle-node bifurcation point of equilibrium point systems, adequate convergence can be achieved even if the error from the true eigenvalue is around 10% (or more).
• For example, for the saddle-node bifurcation point of a limit cycle or a periodic solution, 1% to few % error is considered appropriate. However, it may depend largely on the problem. Some trial and error may be necessary.

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.

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.
• 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 In default settings, this terminated mode is selected.
*2 For the meaning of iter, gmax, and emax, see for example, the FIX manual: parameters for Newton's methods.
*3 In almost all cases, it may not need to change these values.

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