About FIX options
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:
The procedure to terminate FIX can be configured using the following steps.
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¡Çs 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¡Çs method is inaccurate, it may give rise to problems such as the following:
In the default settings, when FIX is initially started, the convergence precision is set to 1E6. 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¡Çs method.
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¡Çs method*2:
In particular, if the Newton¡Çs method does not converge in the number of iterations iter, Newton¡Çs 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:
The following steps are used to change the divergence criteria for Newton¡Çs method.
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¡Çs method. Approximation using a 5thorder 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.
How to change the detection precision of bifurcation point
The parameter value where Newton¡Çs 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¡Çs 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 ¦Ì, the program stops when ¡¡¡Ã¦Ì*  ¦Ì¡Ã< 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.
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¡Çs 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.
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¡Çs 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.
