About FIX operations
- Common
- Autonomous system
- Non-autonomous system
- Discrete system
How to pass information of the detected bifurcation point to BF
After a bifurcation point is detected, information such as the values of the state variables and parameters must be passed to BF. Of course the data may be manually input in one by one. However, mistakes may happen, and this is a cumbersome job when there are many variables.
As one method, the information when a bifurcation point is detected is saved to a temporary file. After that, the temporary file is imported by BF. To do so, the following steps can be used:
- Detect of a bifurcation point.
- Select Tools -> Export BF point.
- Export the information of the detected bifurcation point. the exported information are displayed in the list box.
- Select Program -> BF.
- The following figure shows the state when BF starts up at first time.
- Select BF: Tools -> Import BF Point.
- The information of the detected bifurcation point are imported to each field.
How to restart FIX after the stop button is clicked
While searching for bifurcation points in FIX, you may want to:
- Change parameters that are changed for searching bifurcations.
- Change the step size of the parameter that is changed.
- Change the configuration options.
In these cases, calculations have to be stopped by clicking the Stop Button. Then the configurations are changed and calculations are restarted. However, clearing all the results calculated prior to stopping and recalculating is a waste of time and resources. Therefore, steps are shown below how to resume the calculation with parameters when the calculation is stopped.
- Start search by clicking the Start Button.
- Terminate the calculation to click the Stop Button.
- Right click on mouse in the list box, and select Import current states.
- The values of state variables and parameters when the calculations are terminated are reflected in the respective fields.
- After the parameters when the calculations are terminated are reflected in the respective fields, change the parameters that are changed for searching bifurcations or step size, and click the Start Button again. The calculations (search for bifurcation point) will restart.
How to search for bifurcations of an equilibrium point
When a stable equilibrium point is found with PP, we should consider searching the bifurcation point of this equilibrium point. The following steps are used.
- Import the initial condition from PP. See How to import initial conditions to run FIX.
- Check whether the analysis mode is the equilibrium point (Equilibrium).
- Select the parameters to change. Decide in which direction to change the parameters on the two-parameter plane. X and Y show the parameters that were selected for the x-
and y-axis.
- Set the step size according to the change in parameters. In this example, the parameters are only changed in the x-axis direction. Of course, it is possible to change the parameters in the y-axis direction only, or diagonally on the parameter plane.
- 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.
- Click the Start Button to start the calculation.
- Evaluate the eigenvalues of the equilibrium point by making minute changes in the parameters. Here, a new window will start. This window represents a Gauss plane, and the positions of the current eigenvalues are shown. The following example shows the detection of a Hopf bifurcation point when an externally applied DC current is increased in the Hodgkin-Huxley equation.
- Confirm that this is a Hopf bifurcation. In the figure below, the real part of the eigenvalues closely approaches zero, and there is a non-zero imaginary part to the eigenvalues, and therefore the eigenvalues are purely imaginary numbers. This means the parameters for bifurcation occurrence, and the coordinates of the equilibrium point are obtained. Thus, an initial bifurcation point was obtained.
- If an initial bifurcation point is obtained, the bifurcation set on a two-parameter plane can be calcurate to pass the information to BF, see How to pass information of the detected bifurcation point to BF.
How to search for bifurcation of limit cycles
When a stable limit cycle is found with PP, we should consider searching the bifurcation point of this limit cycle. The following steps are used.
- Import the initial condition from PP. See How to import initial conditions to run FIX.
- Check whether the analysis mode is the Limit cycle (Limit cycle).
- Check whether the state variables selected as the Poincare section configured in PP and the return time (period of limit cycle) accurately reflects the state in PP.
- Select the parameters to change. Decide in which direction to change the parameters on the two-parameter plane. X and Y show the parameters that were selected for the x-
and y-axis.
- Set the step size according to the change in parameters. In this example, the parameters are only changed in the x-axis direction. Of course, it is possible to change the parameters in the y-axis direction only, or diagonally on the parameter plane.
- 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.
- Click the Start Button to restart the calculation.
- Evaluate the characteristic multipliers (eigenvalues) of the limit cycle by making minute changes in the parameters. Here, a new window will start. This window represents a Gauss plane, and the positions of the current characteristic multipliers are shown. The following example shows the detection of a Saddle-node bifurcation point when an externally applied DC current is increased in the Morris-Lecar neuron model.
- Confirm that this is a saddle-node bifurcation of a limit cycle. The following figure shows the positions of characteristic multipliers on the Gauss plane. The anlaysis of a limit cycle is done by considering a Poincare section and reducing a continuous orbit of the limit cycle to the movement of points on the Poincare section. Here, one of the characteristic multipliers is always 1. This reflects the fact that the orbit of the limit cycle transversally crosses with the Poincare section. We see that the real part of one of the remaining characteristic multiplier approaches "£±" , but never becomes exactly 1. This is because the Jacobian matrix in the Newton”Ēs method used by FIX becomes a non-singular matrix at the saddle-node bifurcation point, and Newton”Ēs method itself does not converge. If the characteristic multiplier is close enough to "£±", we have obtained a good approximation of the bifurcation point.
- If an initial bifurcation point is obtained, the bifurcation set on a two-parameter plane can be calcurate to pass the information to BF, see How to pass information of the detected bifurcation point to BF.
How to search for bifurcation of periodic solutions (fixed or periodic points)
When a stable periodic solution is found with PP, we should consider searching the bifurcation point of this periodic solution. The following steps are used.
- Import the initial condition from PP. See How to import initial conditions to run FIX.
- Check whether the period of the Poincare map configured in PP accurately reflects the state in PP.
- Select the parameters to change. Decide in which direction to change the parameters on the two-parameter plane. X and Y show the parameters that were selected for the x-
and y-axis.
- Set the step size according to the change in parameters. In this example, the parameters are only changed in the x-axis direction. Of course, it is possible to change the parameters in the y-axis direction only, or diagonally on the parameter plane.
- 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.
- Click the Start Button to restart the calculation.
- 1.Evaluate the characteristic multipliers of the periodic point by making minute changes in the parameters. Here, a new window will start. This window represents a Gauss plane, and the positions of the current characteristic multipliers are shown. When the frequency of the periodic external force (omega) is increased, two complex conjugate characteristic multipliers approach "|1|" and cross the unit circle. Here, it means that the Neimark-Sacker bifurcation occured. In the default settings, switch III is selected as shown in How to change the action when detects a bifurcation point. Therefore, when FIX detects that the Neimark-Sacker bifurcation point is passed, the direction of the parameter change is reversed and calculation starts backwards. Every time the bifurcation point is passed, the step size is halved to narrow in the parameters that cause bifurcation. The FIX stops after an accurate bifurcation point is obtained. In the following example, the Neimark-Sacker bifurcation for periodic oscillations observed in a circadian oscillator model in Neurospora is detected.
- When a bifurcation point is detected, the values of the characteristic multipliers should be checked. See how the absolute values of the two characteristic multipliers approach 1. If the absolute values of the two characteristic multipliers are close enough to "£±", we have obtained a good approximation of the Neimark-Sacker bifurcation point.
- If an initial bifurcation point is obtained, the bifurcation set on a two-parameter plane can be calcurate to pass the information to BF, see How to pass information of the detected bifurcation point to BF.
How to search for bifurcation of periodic points (fixed points)
When a stable periodic point is found with PP, we should consider searching the bifurcation point of this periodic point. The following steps are used.
- Import the initial condition from PP. See How to import initial conditions to run FIX.
- Check whether the period of the Poincare map configured in PP accurately reflects the state in PP.
- Select the parameters to change. Decide in which direction to change the parameters on the two-parameter plane. X and Y show the parameters that were selected for the x-
and y-axis.
- Set the step size according to the change in parameters. In this example, the parameters are only changed in the x-axis direction. Of course, it is possible to change the parameters in the y-axis direction only, or diagonally on the parameter plane.
- 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.
- Click the Start Button to restart the calculation.
- Evaluate the characteristic multipliers of the periodic point by making minute changes in the parameters. Here, a new window will tart. This window represents a Gauss plane, and the positions of the current characteristic multipliers are shown. When the parameter is increased, one of the characteristic multipliers approach "-1" and passes –1. Here, it means that the period-doubling bifurcation occured. In the default settings, switch III is selected as shown in How to change the action when detects a bifurcation point. Therefore, when FIX detects that the period-doubling bifurcation point is passed, the direction of the parameter changeis reversed and calculation starts backwards. Every time the bifurcation point is passed, the step size is halved to narrow in the parameters that cause bifurcation. FIX stops after an accurate bifurcation point is obtained. In the following example, the occurrence of the period-doubling bifurcation of a 5-periodic point ovserved in the Henon map is shown.
- When a bifurcation point is detected, the values of the characteristic multipliers should be checked. See how the absolute value of a characteristic multipliers approach -1. If the value of the characteristic multiplier is close enough to -1, we have obtained a good approximation of the bifurcation point.
- If an initial bifurcation point is obtained, the bifurcation set on a two-parameter plane can be calcurate to pass the information to BF, see How to pass information of the detected bifurcation point to BF.