BF (Bifurcation Curve Tracker)

What's this ?

This tool is used to trace bifurcation curves on an arbitrary parameter plane. Based on the bifurcation parameter values obtained by the bifurcation tool FIX, bifurcation parameter values are obtained by repetitively changing one parameter minutely and applying the Newton method. An invariant set in which the bifurcation occurs can be obtained by this algorithm.


Main Panel

  • Mode : Types of Bifurcation.

    • Equilibrium : Used to obtain bifurcation sets of an equilibrium point
      • Hopf : For the Hopf bifurcation sets
      • SN:For the Saddle-node bifurcation sets
      • Pf:For the Pitch-fork bifurcation sets


    • Periodic : Use to obtain bifurcation sets of a limit cycle or a periodic solution.
      • Pf:For the Pitch-fork bifurcation sets
      • NS:For the Neimark-Sacker bifurcation sets
      • SN:For the Saddle-node bifurcation sets
      • PD:For the Period-doubling bifurcation sets


  • Initial states
    • Configuration of State variables and the other parameters
      • x[1], x[2], ... , x[n] : Each value of the state variables
      • Section: State variables set as the Poincare section (Only for the limit cycle observed in the autonomous system)
      • Period : Period of a periodic point in Poincare map (Limit cycle of an autonomous system, a non-autonomous system, a discrete system)
      • Return time : Return time of a limit cycle (Only for the limit cycle of an autonomous system)
      • omega(theta) : The information of an imaginary part or an argument of the complex conjugate eigenvalues is input to calculate the Hopf or the NS bifurcation. (Only for the Hopf and NS calculations)

        (a-1) The case of the equilibrium point of an Autonomous system.

        (a-2) The case of the limit cycle of an Autonomous system.

        (b) The case of a non-autonomous system.

        (c) The case of a discrete system.

  • Parameters
    • Configuration of System Parameters and the other parameters
      • X : X-axis parameter in 2-parameter bifurcation diagram. When starting calculations, it is treated as the unknown variable parameter of the Newton method.
      • Y ; Y-axis parameter in 2-parameter bifurcation diagram. When starting calculations, it is treated as the control variable.
      • start : The values of system parameters when BF starts. If initial values are imported from FIX, these values are reflected to each field.
      • step size : Step size for changing parameters. Displayed when X or Y is selected*1
      • stop : Stop values of the changing parameter. When system parameters reach this value, the calculation stops.


  • Output file
    • File name : Outputs an information, e.g., the value of system parameters, state variables, eigenvalues, and so on, displayed in the list box to a file (Default configuration is ON). This output file is therefore Bifurcation set data.


  • Start, Stop
    • Controls when to start/stop calculations.


  • ODE Solver(Limited to the autonomous and the non-autonomous systems)
    • Displays the name of the Solver used to conduct numerical integration. For details refer to Setting.


  • List Box
    • Displays the calculation results. The numbers displayed correspond to the following respectively:
      Number of iterations) para(X) para(Y) / x[1], x[2], ・・・, x[n] |
      Re(Eigenvalue), Im(Eigenvalue), absolute value |・・・ |・・・ |(Jacobian)


    • It is possible to reflect the current condition in the respective item fields by right-clicking in this list box*2.

Menu Bar

  • File
    • clear : If selected, clears to the configuration when BF was started up.~

  • Import : Imports previously saved BF configurations and initial condition files*3.
    • BF save file : Imports previously saved BF configurations (XML file)
    • Text data : Imports contents of a text file into Initial states and Parameters fields (text file)


  • Export : Exports all current BF configurations, or exports initial values and parameters into a text file*4.
    • BF save file : Exports current BF settings (XML file)
    • Text data : Exports data in Initial states and Parameters fields as text data (text file)


  • Close : Quit BF


  • Program
    • SE : Starts the SE panel for the current Project.
    • PP : Starts the PP tool for the current Project.
    • FIX : Starts the FIX tool for the current Project.
    • BF : Starts the BF tool for the current Project.
    • BFP : Starts the tool for making a bifurcation diagram.


  • Project
    • Change Project: This is used to generate a project other than the current generated project, or to load already existing projects.


  • Tools
    • Import BF point : Used to reflect parameters and state variables when bifurcation occurs that was obtained by FIX *5.


  • Help
    • Manual : Startup the manual of BF.
      • Select Japanese or English

        (a) Selection of the Manual

        (b) Start up the BF manual

    • about BunKi : A brief introduction of BunKi.
      • Version information, contact lists to developer team, credit, etc.

        (a) Selected about BunKi

        (b) Display the credit for BunKi

*1 The basic control direction follows the step size of the parameter selected at the field Y. However, the control parameter switches between X and Y according to the calculation result. So, users also need to set the step size related to the field X.
*2 For details, refer to How to restart BF from a state that the calculation is stopped
*3 For details, refer to How to import an initial setting for BF and How to import an initial value from a text file.
*4 For details, refer to How to save the configuration of BF ?, and How to create a text file consisting of initial values and parameter values.
*5 For details, refer to How to import an initial value to start BF.

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Last-modified: 2009-07-23 (Thu) 20:13:30 (3229d)