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.

• 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.

 gmax If the sum of all the absolute value of the variables exceeds this value, the program considers it diverged. emax If the sum of all the absolute value of the system variables used in the Newton method exceeds this value, the program considers it diverged. eps If 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. feps If 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. iter Number of convergences for the Newton method. One exceeded this number, the program halts.

Step size control

• 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.

• 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.

• On : Use a variable step size
• Off : Use a fixed step size(Default setting)*1
• 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 periodic points and the detailed settings

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

.

• 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 periodic points with a permutation symmetry. The periodic point is an invariant under the operation by the following matrix:

,

where indicates the following matrix:

.

• 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 the periodic point with an arbitrary symmetrical property. User must define the arbitrary symmetrical matrix.*2
• divided numberNot use for Discrete systems

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.

*1 We strongly recommend to set off on this option at this time.
*2 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.

Last-modified: 2009-07-23 (Thu) 20:13:30 (3650d)