About BF operation




What if the bifurcation curve that is calculated at real time is shown out of the window range ?

When the bifurcation calculation starts, the calculated results of the bifurcation set is shown in real time in another window. However, if calculations continue out of the range of the window, the curve will not be shown. In such a case, these steps are used to rescale the range such that the curve is displayed.

  1. When BF starts, a different drawing window starts as the bifurcation diagram.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFfig.png

  2. Select Pan tool.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFfig2.png

  3. When the calculated results move out of the range of the drawing window, right click in the drawing window. The following can be selected:
    • Reset to Original Display
    • Pan Options

      index.php?plugin=attach&refer=aboutBF_operations&openfile=BFfig3.png

  4. When Reset to Original Display is selected, MATLAB rescales such that the current calculation results fit in the window.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFfig4.png



How to obtain the saddle-node bifurcation set of an equilibrium point

After detecting a saddle-node bifurcation point of an equilibrium point using FIX, the bifurcation set of the equilibrium point is calculated in an arbitrary two-parameter plane using BF.

  1. After importing the initial bifurcation point (see How to import initial conditions to run BF),check whether Equilibrium: SN is checked as the mode in the BF main panel.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqsn.png

  2. Select two parameters to decide on which two-parameter planes to calculate the bifurcation set. By selecting the X and Y parameters in Parameters field, the two selected parameters are assigned.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqsn2.png

  3. Set the step size for parameter change. If stop values of parameters are to be configured, change the Default value in the stop field. Then input the stop value of the parameters.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqsn3.png

  4. Input the file name to save the calculation results. In the default settings, the data is saved to a file named bf.out. If a saved file with the same name exists, BF asks whether to overwrite the file when starting calculations.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqsn4.png

  5. Click the Start button to start the calculation.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqsn5.png

  6. In the list box in the main panel, the values of parameters and the state variables, eigenvalue information, and other information are displayed in the following order. From right to left:
    1. iteration): Number of iterations until convergence in Newton's method
    2. parameter1, parameter2 / : The values of X and Y parameters
    3. x[1], x[2], ... , x[n] | : The values of state variables
    4. real[1] imag[1] abs[1] | real[2] imag[2] abs[2] | ... | real[n] imag[n] abs[n] : eigenvalues. The eigenvalues are displayed in the following order: real part, imaginary part, and absolute value. For an n-dimensional system, information for n eigenvalues is displayed.
    5. (Jacobian) : The value of the determinant of the Jacobian matrix used in Newton's method.
    6. Period : Equilibrium point have no period, and dtherefore 0 is always displayed.

      index.php?plugin=attach&refer=aboutBF_operations&openfile=eqsn6.png



How to obtain the Hopf bifurcation set of an equilibrium point

After detecting a Hopf bifurcation point of an equilibrium point using FIX, the bifurcation set of the equilibrium point is calculated in an arbitrary two-parameter plane using BF.

  1. After importing the initial bifurcation point (see How to import initial conditions to run BF),check whether Equilibrium: Hopf is checked as the mode in the BF main panel.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqhp.png

  2. For a calculation of the Hopf bifurcation set, the values of the purely imaginary eigenvalues when bifurcation occurs are necessary. When the bifurcation point is obtained with FIX, confirm that there are a pair of complex conjugate, purely imaginary eigenvalues in the eigenvalues, and check the values. The values should be shown in the Omega field.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqhp2.png

  3. Select two parameters to decide on which two-parameter planes to calculate the bifurcation set. By selecting the X and Y parameters in Parameters field, the two selected parameters are assigned.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqhp3.png

  4. Set the step size for parameter change. If stop values of parameters are to be configured, change the Default value in the stop field. Then input the stop value of the parameters.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqhp4.png

  5. Input the file name to save the calculation results. In the default settings, the data is saved to a file named bf.out. If a saved file with the same name exists, BF asks whether to overwrite the file when starting calculations.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqhp5.png

  6. Click the Start button to start the calculation.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqhp6.png

  7. In the list box in the main panel, the values of parameters and the state variables, eigenvalue information, and other information are displayed in the following order. From right to left:
    1. iteration): Number of iterations until convergence in Newton's method
    2. parameter1, parameter2 / : The values of X and Y parameters
    3. x[1], x[2], ... , x[n] | : The values of state variables
    4. real[1] imag[1] abs[1] | real[2] imag[2] abs[2] | ... | real[n] imag[n] abs[n] : eigenvalues. The eigenvalues are displayed in the following order: real part, imaginary part, and absolute value. For an n-dimensional system, information for n eigenvalues is displayed.
    5. (Jacobian) : The value of the determinant of the Jacobian matrix used in Newton's method.
    6. Period : Equilibrium point have no period, and dtherefore 0 is always displayed.

      index.php?plugin=attach&refer=aboutBF_operations&openfile=eqhp7.png



How to obtain the pitch-fork bifurcation set of an equilibrium point

After detecting a pitch-fork bifurcation point of an equilibrium point using FIX, the bifurcation set of the equilibrium point is calculated in an arbitrary two-parameter plane using BF.

  1. After importing the initial bifurcation point (see How to import initial conditions to run BF),check whether Equilibrium: Pf is checked as the mode in the BF main panel.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqpf.png

  2. Select two parameters to decide on which two-parameter planes to calculate the bifurcation set. By selecting the X and Y parameters in Parameters field, the two selected parameters are assigned.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqpf2.png

  3. Set the step size for parameter change. If stop values of parameters are to be configured, change the Default value in the stop field. Then input the stop value of the parameters.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqpf3.png

  4. Input the file name to save the calculation results. In the default settings, the data is saved to a file named bf.out. If a saved file with the same name exists, BF asks whether to overwrite the file when starting calculations.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqpf4.png

  5. Click the Start button to start the calculation.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=eqpf5.png

  6. In the list box in the main panel, the values of parameters and the state variables, eigenvalue information, and other information are displayed in the following order. From right to left:
    1. iteration): Number of iterations until convergence in Newton's method
    2. parameter1, parameter2 / : The values of X and Y parameters
    3. x[1], x[2], ... , x[n] | : The values of state variables
    4. real[1] imag[1] abs[1] | real[2] imag[2] abs[2] | ... | real[n] imag[n] abs[n] : eigenvalues. The eigenvalues are displayed in the following order: real part, imaginary part, and absolute value. For an n-dimensional system, information for n eigenvalues is displayed.
    5. (Jacobian) : The value of the determinant of the Jacobian matrix used in Newton's method.
    6. Period : Equilibrium point have no period, and dtherefore 0 is always displayed.

      index.php?plugin=attach&refer=aboutBF_operations&openfile=eqpf6.png



How to obtain the saddle-node bifurcation set of limit cycle / periodic solution / periodic point

After detecting a saddle-node bifurcation point of a limit cycle(autonomous system), a periodic solution(non-autonomous system), and a periodic point(discrete system) using FIX, the bifurcation set of the equilibrium point is calculated in an arbitrary two-parameter plane using BF.

  1. After importing the initial bifurcation point (see How to import initial conditions to run BF),check whether Periodic: SN is checked as the mode in the BF main panel.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFsn.png

  2. Select two parameters to decide on which two-parameter planes to calculate the bifurcation set. By selecting the X and Y parameters in Parameters field, the two selected parameters are assigned.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFsn2.png

  3. Set the step size for parameter change. If stop values of parameters are to be configured, change the Default value in the stop field. Then input the stop value of the parameters.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFsn3.png

  4. Input the file name to save the calculation results. In the default settings, the data is saved to a file named bf.out. If a saved file with the same name exists, BF asks whether to overwrite the file when starting calculations.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFsn4.png

  5. Click the Start button to start the calculation.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFsn5.png

  6. In the list box in the main panel, the values of parameters and the state variables, eigenvalue information, and other information are displayed in the following order. From right to left:
    1. iteration): Number of iterations until convergence in Newton's method
    2. parameter1, parameter2 / : The values of X and Y parameters
    3. x[1], x[2], ... , x[n] | : The values of state variables
    4. real[1] imag[1] abs[1] | real[2] imag[2] abs[2] | ... | real[n] imag[n] abs[n] : eigenvalues. The eigenvalues are displayed in the following order: real part, imaginary part, and absolute value. For an n-dimensional system, information for n eigenvalues is displayed.
    5. (Jacobian) : The value of the determinant of the Jacobian matrix used in Newton's method.
    6. Period : depends on the dynamical system. The following information is shown:
      1. autonomous system : Return time of the limit cycle.
      2. non-autonomous system : Time between stroboscopic maps
      3. discrete system : The period not displayed.
        index.php?plugin=attach&refer=aboutBF_operations&openfile=BFsn6.png



How to obtain the period-doubling bifurcation set of limit cycle / periodic solution / periodic point

After detecting a period-doubling bifurcation point of a limit cycle(autonomous system), a periodic solution(non-autonomous system), and a periodic point(discrete system) using FIX, the bifurcation set of the equilibrium point is calculated in an arbitrary two-parameter plane using BF.

  1. After importing the initial bifurcation point (see How to import initial conditions to run BF),check whether Periodic: PD is checked as the mode in the BF main panel.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFpd.png

  2. Select two parameters to decide on which two-parameter planes to calculate the bifurcation set. By selecting the X and Y parameters in Parameters field, the two selected parameters are assigned.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFpd2.png

  3. Set the step size for parameter change. If stop values of parameters are to be configured, change the Default value in the stop field. Then input the stop value of the parameters.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFpd3.png

  4. Input the file name to save the calculation results. In the default settings, the data is saved to a file named bf.out. If a saved file with the same name exists, BF asks whether to overwrite the file when starting calculations.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFpd4.png

  5. Click the Start button to start the calculation.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFpd5.png

  6. In the list box in the main panel, the values of parameters and the state variables, eigenvalue information, and other information are displayed in the following order. From right to left:
    1. iteration): Number of iterations until convergence in Newton's method
    2. parameter1, parameter2 / : The values of X and Y parameters
    3. x[1], x[2], ... , x[n] | : The values of state variables
    4. real[1] imag[1] abs[1] | real[2] imag[2] abs[2] | ... | real[n] imag[n] abs[n] : eigenvalues. The eigenvalues are displayed in the following order: real part, imaginary part, and absolute value. For an n-dimensional system, information for n eigenvalues is displayed.
    5. (Jacobian) : The value of the determinant of the Jacobian matrix used in Newton's method.
    6. Period : depends on the dynamical system. The following information is shown:
      1. autonomous system : Return time of the limit cycle.
      2. non-autonomous system : Time between stroboscopic maps
      3. discrete system : The period not displayed.
        index.php?plugin=attach&refer=aboutBF_operations&openfile=BFpd6.png



How to obtain the Neimark-Sacker bifurcation set of limit cycle / periodic solution / periodic point

After detecting a Neimark-Sacker bifurcation point of a limit cycle(autonomous system), a periodic solution(non-autonomous system), and a periodic point(discrete system) using FIX, the bifurcation set of the equilibrium point is calculated in an arbitrary two-parameter plane using BF.

  1. After importing the initial bifurcation point (see How to import initial conditions to run BF),check whether Periodic: NS is checked as the mode in the BF main panel.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns.png

  2. The Neimark-Sacker bifurcation set is calculated by not using only fixed point conditions and characteristic equations, but also using information on the argument of the characteristic multiplier when bifurcation occurs. Therefore, it is necessary to also obtain information on the argument from the relation between the real and imaginary parts of the characteristic multiplier when bifurcation occurs. When bifucation points are detected using FIX, confirm that a pair of complex conjugate characteristic multipliers with absolute value 1 exists among the characteristic multipliers, and check the value. The information on the argument should be displayed in the theta field.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns2.png

    (a) For an autonomous system

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns3.png

    (b) For a non-autonomous system

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns4.png

    (c) For a discrete system

  3. Select two parameters to decide on which two-parameter planes to calculate the bifurcation set. By selecting the X and Y parameters in Parameters field, the two selected parameters are assigned.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns5.png

  4. Set the step size for parameter change. If stop values of parameters are to be configured, change the Default value in the stop field. Then input the stop value of the parameters.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns6.png

  5. Input the file name to save the calculation results. In the default settings, the data is saved to a file named bf.out. If a saved file with the same name exists, BF asks whether to overwrite the file when starting calculations.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns7.png

  6. Click the Start button to start the calculation.

    index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns8.png

  7. In the list box in the main panel, the values of parameters and the state variables, eigenvalue information, and other information are displayed in the following order. From right to left:
    1. iteration): Number of iterations until convergence in Newton's method
    2. parameter1, parameter2 / : The values of X and Y parameters
    3. x[1], x[2], ... , x[n] | : The values of state variables
    4. real[1] imag[1] abs[1] | real[2] imag[2] abs[2] | ... | real[n] imag[n] abs[n] : eigenvalues. The eigenvalues are displayed in the following order: real part, imaginary part, and absolute value. For an n-dimensional system, information for n eigenvalues is displayed.
    5. (Jacobian) : The value of the determinant of the Jacobian matrix used in Newton's method.
    6. Period : depends on the dynamical system. The following information is shown:
      1. autonomous system : Return time of the limit cycle.
      2. non-autonomous system : Time between stroboscopic maps
      3. discrete system : The period not displayed.
        index.php?plugin=attach&refer=aboutBF_operations&openfile=BFns9.png



How to obtain the pitch-fork bifurcation set of limit cycle / periodic solution / periodic point

Under construction !




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Last-modified: 2009-07-23 (Thu) 20:13:30 (3374d)