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.

    index.php?plugin=attach&refer=FIXatNonautonomous&openfile=detect.png


    Iwhen detecting a bifurcation value, keep on the calculations without halting.
    IIwhen detecting a bifurcation value, halt the program.
    IIIwhen 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 saddle-node bifurcation. This is the parameter that will fix how close to the exact eigenvalue we want to set the precision.

    index.php?plugin=attach&refer=FIXatNonautonomous&openfile=seig.png

    • Saddle-node bifurcation of a fixed/periodic point
      • If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition (index.php?plugin=attach&refer=FIXatNonautonomous&openfile=eq_img01.gif) 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 (index.php?plugin=attach&refer=FIXatNonautonomous&openfile=eq_img02.gif) 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 (index.php?plugin=attach&refer=FIXatNonautonomous&openfile=eq_img03.gif) 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 (index.php?plugin=attach&refer=FIXatNonautonomous&openfile=eq_img01.gif) 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.

        index.php?plugin=attach&refer=FIXatNonautonomous&openfile=figure.png



Newton's method

  • parameters for Newton's methods: The precise position of a fixed/periodic point of the periodic solution is computed by the Newton method. Each of the following items is a control parameter of this Newton method.

    index.php?plugin=attach&refer=FIXatNonautonomous&openfile=newton.png

    gmaxIf the sum of all the absolute value of the derivatives with respect to the variables exceeds this value, the program considers it diverged.
    emaxIf the sum of all the absolute value of the system variables used in the Newton method exceeds this value, the program considers it diverged.
    epsIf 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.
    fepsIf 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.
    iterLimit amount of iterations for the Newton method to converge. If it exceeds this number, the program halts.
    nnnExtrapolating 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.


ODE solver

  • parameters for ODE solver:Selection and detailed settings of numerical integration.
    • Solver:selection of ODE solver built in MATLAB is possible.

      index.php?plugin=attach&refer=FIXatNonautonomous&openfile=solver.png


      ode45This is based on an explicit Runge-Kutta formula, the Dormand-Prince pair. It is a one-step solver; that is, in computing y(tn), it needs only the solution at the immediately preceding time point, y(tn-1). In general, ode45 is the best solver to apply as a "first try" for most problems.
      ode23This is also based on an explicit Runge-Kutta proposed by Bogacki and Shampine. It may be more efficient than ode45 at crude tolerances and in the presence of mild stiffness. ode23 is a one-step solver.
      ode113This is a variable order Adams-Bashforth-Moulton PECE solver. It may be more efficient than ode45 at stringent tolerances. ode113 is a multi-step solver; that is, it normally needs the solutions at several preceding time points to compute the current solution.
      ode23s};This is based on a modified Rosenbrock formula of order 2. Because it is a one-step solver, it may be more efficient than ode15s at crude tolerances. It can solve some kinds of stiff problems for which ode15s is not effective.
      ode15sThis is a variable order solver based on the numerical differentiation formulas (NDFs). These are related to but are more efficient than the backward differentiation formulas, BDFs (also known as Gear's method). Like ode113, ode15s is a multi-step method solver. If you suspect that a problem is stiff or if ode45 failed or was very inefficient, try ode15s.
      myselfThis is a standard fixed step 4th order Runge Kutta method.It is not recommended unless under particular circumstances.

      index.php?plugin=attach&refer=FIXatNonautonomous&openfile=solver.png

    • Initial step : The initial step size that the ODE solver uses. When using one of the pre-installed solvers, since the step size is automatically adjusted, setting an initial step size unreasonably small (enter a value of the input field too big) will result in poor computation efficiency. When selecting the myself method, its step size is simply 1/(the value of the input field).
    • Relative tolerance : measures the error relative to the size of each state. The relative tolerance represents a percentage of the state's value.
    • Absolute tolerance : represents the acceptable error as the value of the measured state approaches zero*1.


Symmetrical property

  • Symmetrical property:Definition of symmetrical properties of periodic solutions and the detailed settings

    index.php?plugin=attach&refer=FIXatNonautonomous&openfile=symmetry.png

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

      index.php?plugin=attach&refer=FIXatNonautonomous&openfile=eq_img04.gif

      for details see Ref.[1].
    • Permutation:The case of periodic solutions with a permutation symmetry and a time shift. The periodic solution is an invariant under the operation by the following matrix and by the time shift:

      index.php?plugin=attach&refer=FIXatNonautonomous&openfile=eq_img05.gif,

      where index.php?plugin=attach&refer=FIXatNonautonomous&openfile=eq_img06.gif indicates the following matrix:

      index.php?plugin=attach&refer=FIXatNonautonomous&openfile=eq_img07.gif,

      for detail see Ref.[1].
    • Arbitrary:The case of periodic solutions with an arbitrary symmetrical property. User must define the arbitrary symmetrical matrix.
      • divided number:When the symmetry operation is carried out to a periodic solution, you must also take into account the time shift([1]).This option is given as a divided number that need to shift how long time for the period of the periodic solution.

References


  • [1] Y. Katsuta and H. Kawakami,Bifurcations of periodic solutions in a nonlinear nonautonomous system with symmetry, Electronics and communications in Japan, Scripta Technica, Inc., Part 3, vol.77, no.3, pp.106--116, 1994.





*1 for detail see odeset

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Last-modified: 2009-07-23 (Thu) 20:13:30 (3310d)