LEFT:[[FIX>FIX_en]]
CENTER:[[manual>manual_en]]
RIGHT:[[FrontPage]]~
&br;&br;

RIGHT:&size(30){FIX Options};~
&br;&br;

-&size(20){[[Detection>FIXatAutonomous_en#detect]]};
--&size(15){[[detection method of bifurcations>FIXatAutonomous_en#detect]]};
--&size(15){[[SeigTol val>FIXatAutonomous_en#eigtol]]};
-&size(20){[[Graphic display>FIXatAutonomous_en#iograph]]};
-&size(20){[[parameters for Newton's methods>FIXatAutonomous_en#newtonpara]]};
--&size(15){[[parameters for Newton's methods>FIXatAutonomous_en#newtonpara]]};
-&size(20){[[ODE solver>FIXatAutonomous_en#solver]]};
--&size(15){[[parameters for ODE solver>FIXatAutonomous_en#solver]]};
-&size(20){[[Symmetrical property>FIXatAutonomous_en#symmetry]]};~
&br;&br;

CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=all.png);~

&br;&br;
&aname(detect);
LEFT:&size(20){&color(green){''Detection''};};
----
-&size(15){&color(green){''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.};~
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=detect.png);~
&br;&br;
|&color(blue){''I''};| when detecting a bifurcation value, keep on the calculations without halting.|
|&color(blue){''II''};| when detecting a bifurcation value, halt the program.|
|&color(blue){''III''};| when detecting a bifurcation value, adjust the step size for a variational parameter and automatically compute the closest values when the bifurcation occurs.|
&br;&br;

&aname(eigtol);
-&size(15){&color(green){''SeigTol_val''}; : If you selected the above detection method "&color(blue){''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.};~
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=seig.png);~
&br;
--''Saddle-node bifurcation'' of a &color(magenta){equilibrium point};
---If the distance between the current eigenvalue and the bifurcation condition (&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img01.gif);) reached within the range of the SeigTol value, the program halts.
--''Hopf bifurcation'' of a &color(magenta){equilibriumpoint};
---If the distance between the current eigenvalue and the bifurcation condition (&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img02.gif); and &ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img03.gif);) reached within the range of the SeigTol value, the program halts.
--''Saddle-node bifurcation'' of a &color(magenta){limit cycle};
---If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition (&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img04.gif);) reached within the range of the SeigTol value, the program halts.
--''Period-doubling bifurcation'' of a &color(magenta){limit cycle};
---If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition (&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img05.gif);) reached within the range of the SeigTol value, the program halts.
--''Neimaark-Sacker bifurcation'' of a &color(magenta){limit cycle};
--''Neimark-Sacker bifurcation'' of a &color(magenta){limit cycle};
---If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition (&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img06.gif);) reached within the range of the SeigTol value, the program halts.
--''Pitch-fork bifurcation'' of a &color(magenta){limit cycle};
---If the distance between the current characteristic multiplier (eigenvalue) and the bifurcation condition (&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img04.gif);) reached within the range of the SeigTol value, the program halts.~


&br;&br;
&aname(iograph);
LEFT:&size(20){&color(green){''Graphic display''};};
----
-&size(15){&color(green){''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 &color(red){''one''}; window for all computations.
---''Off'' : Create a new window every computation. To be used for results comparison, for instance.~
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=figure.png);~
&br;

&br;&br;
&aname(newtonpara);
LEFT:&size(20){&color(green){''Newton's method''};};
----
-&size(15){&color(green){''parameters for Newton's methods''};: The precise position of an equilibrium and a fixed/periodic point of the limit cycle is computed by the Newton method. Each of the following items is a control parameter of this Newton method.};~
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=newton.png);~
&br;
|&color(magenta){''gmax''};| If the sum of all the absolute value of the derivatives with respect to the variables exceeds this value, the program considers it ''diverged''.|
|&color(magenta){''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''.|
|&color(magenta){''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''.|
|&color(magenta){''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''.|
|&color(magenta){''iter''};| Limit amount of iterations for the Newton method to converge. If it exceeds this number, the program halts.|
|&color(magenta){''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.|
&br;&br;

&aname(solver);
LEFT:&size(20){&color(green){''ODE solver''};};
----
-&size(15){&color(green){''parameters for ODE solver''};: Selection and detailed settings of numerical integration.};
--&color(magenta){''Solver''};:selection of [[ODE solver>http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ode23.html]] built in MATLAB is possible.~
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=solver.png);~
&br;&br;
|''[[ode45>http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ode45.html]]''| This 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.|
|''[[ode23>http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ode23.html]]''| This 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.|
|''[[ode113>http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ode113.html]]''| This 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>http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ode23s.html]]''};| 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. |
|''[[ode15s>http://www.mathworks.com/access/helpdesk/help/techdoc/ref/ode15s.html]]''| This 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. |
|''myself''| This is a standard fixed step 4th order Runge Kutta method.It is not recommended unless under particular circumstances.|
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=solver.png);~
&br;
--&color(magenta){''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)''.
--&aname(rel);&color(magenta){''Relative tolerance''}; : measures the error relative to the size of each state. The relative tolerance represents a percentage of the state's value.
--&aname(abs);&color(magenta){''Absolute tolerance''}; : represents the acceptable error as the value of the measured state approaches zero((for detail see [[odeset>http://www.mathworks.com/access/helpdesk/help/techdoc/ref/odeset.html]])).~
&br;&br;

&aname(symmetry);
LEFT:&size(20){&color(green){''Symmetrical property''};};
----
-&size(15){&color(green){''Symmetrical property''};:Definition of symmetrical properties of limit cycles and the detailed settings};~

&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=symmetry.png);~
&br;
--&color(magenta){''Not use this option''};: Out of consideration of a symmetrical property of limit cycles, i.e., ''normal action of FIX''.
--&aname(rel);&color(magenta){''Inversion''};:The case of limit cycles with an inversion symmetry. Thus, the limit cycle have the following property:~
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img07.gif);~
&br;
LEFT:for details see Ref.[[[1>FIXatAutonomous_en#ref1]]].
--&color(magenta){''Permutation''};:The case of limit cycles with a permutation symmetry and a time shift. The limit cycle is an invariant under the operation by the following matrix and by the time shift:~
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img08.gif);,~
&br;
LEFT:where &ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img09.gif); indicates the following matrix:~
&br;
CENTER:&ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=FIXatAutonomous&openfile=eq_img10.gif);,~
&br;
LEFT:for detail see Ref.[[[1>FIXatAutonomous_en#ref1]]].
--&color(magenta){''Arbitrary''};:The case of limit cycle with an arbitrary symmetrical property. User must define the arbitrary symmetrical matrix.
---&color(magenta){''divided number''};:When the symmetry operation is carried out to a limit cycle, you must also take into account the time shift([[[1>FIXatAutonomous_en#ref1]]]).This option is given as a divided number that need to shift how long time for the period of the limit cycle.~
&br;

&size(20){''References''};
----
-&aname(ref1);[1] Y. Katsuta and H. Kawakami, ''Bifurcations of equilibria and periodic solutions in a nonlinear autonomous system with symmetry'', Electronics and Communications in Japan, Scripta Technica, Inc., Part 3, vol.76, no.7, pp.1--14, 1993.~

&br;&br;
LEFT:[[FIX>FIX_en]]
CENTER:[[manual>manual_en]]
RIGHT:[[FrontPage]]

Front page   Edit Diff Backup Upload Copy Rename Reload   New List of pages Search Recent changes   Help   RSS of recent changes