FIX Options

Detection

• detection method of bifurcations : When the value of a system parameter varies, a situation that passes through a bifurcation value may occur. These are the options to control the program's behavior.

 I when detecting a bifurcation value, keep on the calculations without halting. II when detecting a bifurcation value, halt the program. III when detecting a bifurcation value, adjust the step size for a variational parameter and automatically compute the closest values when the bifurcation occurs.

• SeigTol_val : If you selected the above detection method "III", the parameters are automatically adjusted based on the eigenvalues conditioning the bifurcation. This is the parameter that will fix how close to the exact eigenvalue we want to set the precision.

• Saddle-node bifurcation of a fixed/periodic point
• If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition () reached within the range of the SeigTol value, the program halts.
• Period-doubling bifurcation of a fixed/periodic point
• If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition () reached within the range of the SeigTol value, the program halts.
• Neimark-Sacker bifurcation of a fixed/periodic point
• If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition () reached within the range of the SeigTol value, the program halts.
• Pitch-fork bifurcation of a fixed/periodic point
• If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition () reached within the range of the SeigTol value, the program halts.

Graphic display

• Graphic display : The switch to display eigenvalues obtained from the characteristic equation.
• On : display the graph while computing
• Off : do not display the graph while computing
• single figure : see below.
• On : Display the results on one window for all computations.
• Off : Create a new window every computation. To be used for results comparison, for instance.

Newton's method

• parameters for Newton's methods : The precise position of a fixed/periodic point is computed by the Newton 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 Limit amount of iterations for the Newton method to converge. If it exceeds this number, the program halts. nnn Extrapolating the curve already computed yields initial values greatly accelerating the Newton method's convergence speed. But such method cannot be used for the 2 first points of the curve. For the first and the second points, parameters must be partitioned. nnn controls the number of partitions.

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 FIX.
• Inversion：The case of periodic points with an inversion symmetry. Thus, the periodic point have the following property:

• 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:

,

• Arbitrary：The case of periodic points with an arbitrary symmetrical property. User must define the arbitrary symmetrical matrix.
• divided numberNot use for Discrete systems

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