About PP poincare
How to configure the Poincare section
Analyses of limit cycles can be reduced to that of a fixed point of the Poincare map to define the Poincare section. The Poincare section must intersect with the limit cycle. Therefore, the user must configure a Poincare section that intersects the limit cycle based upon the user”Ēs understanding.
There are three methods to configure the Poincare section.
- Select arbitrary state variables and configure the cross-section with arbitrary values
- Select arbitrary state variables and configure the cross-section with coordinates of equilibrium points
- Configure the cross-section based on an arbitrary function
The configuration steps with these methods are as follows.
Configure with an arbitrary fixed value
For example, a repetitive firing can be ovserved in the Hodgkin-Huxley equation when the externally applied DC current increases. To investigate the bifurcation of this limit cycle, it is necessary to obtain points on a Poincare section. Therefore a Poincare section needs to be configured. Here, the user configures arbitrary values of state variables such that the Poincare section intersects with the limit cycle. For example, the section may be configured where the membrane potential crosses 0 [mV].
- Select Setting -> Poincare.
- Start up Configuration window.
- Select Fixed value as the configuration mode.
- Decide where to configure the Poincare section on the limit cycle. Select state variable to configure as the Poincare section.
- Input configured value in text box. In this example, the state variable x[1]( V: corresponds to the membrane potential in the HH equation) was selected, and x[1]=-30 [mV] was chosen as the Poincare section.
- The points on the Poincare section will be shown on the orbit of the limit cycle.
Configure with coordinates of equilibrium point (Part 1)
In general, limit cycles occur when a stable equilibrium point changes to the unstable one. Therefore, the orbit of a limit cycle can be considered to be around the unstable equilibrium point. Thus, by constructing a Poincare section based on the unstable equilibrium point, we can avoid the situation where the orbit of the limit cycle is tangent to the Poincare section while the values of system parameters are changed.
The problem is how to obtain coordinates of the unstable equilibrium point. Proceed with the following steps.
- Select Setting -> Poincare.
- Start up Configuration window.
- Click the search button. Then, start up the input panel of initial values to search an equilibrium point.
- Equilibrium point search starts by giving an appropriate initial value and pressing the calculate button. If the initial value is appropriate, coordinates of an equilibrium point can be obtained*1). If search was successful, click the OK button to apply the coordinates.
(a) Example of a successful search
(b) Example of a failed search
- Check Equilibrium to configure the Poincare section based on obtained coordinates of the equilibrium point.
- From the pull down menu of state variables, select the desired state variable to configure the Poincare section.
- The points on the Poincare section will be shown on the orbit of the limit cycle.
Configure with coordinates of the equilibrium point (Part 2)
If multiple equilibrium points exist near the orbit of the limit cycle, an unwanted equilibrium point may be found when equilibrium points are searched from an arbitrarily determined initial value. Furthermore, the detection of unstable equilibrium points may fail because the initial value was inappropriately given.
In this case, coordinates of the equilibrium point need to be detected by using the FIX tool.
- First, capture a stable equilibrium point with parameters that do not yield a limit cycle. For this procedure, see How to display equilibrium points.
- Next, to investigate the bifurcation of the stable equilibrium point, pass the equilibrium point information to FIX tool. For this procedure, see How to send the attractor information to FIX (part 1) and How to send the attractor information to FIX (part 2).
- Start up FIX tool to search for the occurence point of bifurcations. See How to search for bifurcation of equilibrium points. Here, by using FIX tool, calcurate upon a parameter that a limit cycle can be observed by changing the value of system parametr. It is good if, by using option settings, FIX is configured to a mode*2 that continues calculation after a bifurcation point is detected. Furthermore, set the value in the end parameter field to a value of the system parameter where a limit cycle can be observed.
- The coordinates of the equilibrium point are displayed in the list box when FIX tool stops.
- These coordinates are given to each field as the initial value for searching the equilibrium point. Press the Calculate button to start searching for equilibrium points.
- Finish the search for an unstable equilibrium point. Click the OK button to apply coordinates.
- Check the Equilibrium to configure a Poincare section based on the obtained coordinates of the unstable equilibrium point.
- From the pull down menu of state variables, select the state variable to configure the Poincare section.
- The points on the Poincare section will be shown on the orbit of the limit cycle.
Configure based on arbitrary equation
If the orbit of the limit cycle is symmetric in the phase space, it may be more appropriate to configure a Poincare section in an arbitrary equation. This mode is for such situations*3.
- Select Setting -> Poincare.
- Start up Configuration window.
- Describe the function in the function description field.
- When the equation is input, Equation can be selected as a mode of configuration. Here, select Equation.
- In the following example, the Poincare section is configured using the function x[1]+x[2]=0. The points on the Poincare section will be shown on the orbit of the limit cycle.
How to change the direction when the limit cycle crosses the Poincare section
There are two ways to configure a Poincare section that intersects a limit cycle. The orbit of the limit cycle can cross the Poincare section from above to below, as in the left figure below, or from below to above, as in the right figure below.
Depending on which side of the Poincare section the orbit of a limit cycle intersects, it may affect convergence during numerical calculations. If the convergence is bad in the calculations, maybe changing the direction of intersection with the Poincare section may lead to good results. Here, the method to configure the direction of the intersection of the limit cycle and the Poincare section is shown.
- Select Setting -> Poincare.
- Start up Configuration window.
- Select the direction of the Poincare section from the pull down menu in direction in the Configuration panel.
- +: Configure to intersect with the Poincare section from above (corresponds to left of above figure)
- -: Configure to intersect with the Poincare section from below (corresponds to right of above figure)
- + & -: Configure to intersect with the Poincare section from above and below. However, the Poincare map points used are the values on the section where the orbit intersects from above.
- The following is an example of the Poincare section for different directions. Red points show the Poincare map points on the section.
(a) When
+ is selected.
(b) When
- is selected.
(c) When
+ & - is selected.
How to control the number of the points of the Poincare map (periodic points)
With the default settings of the PP tool, the points of the Poincare map are plotted every time the orbit crosses the section. For example, after starting from a point on the Poincare section, a red point is plotted the next time the orbit crosses the Poincare section. If the orbit of the limit cycle crosses the Poincare section multiple times, with the default settings, the points are plotted in green except the point for the first time the orbit crosses the Poincare section. Therefore, when a Poincare section intersecting with a limit cycle with the period-2 is configured, red and green points are plotted alternately on the orbit drawn in the drawing window. Furthermore, for 2-periodic solutions (or 2-periodic points) in non-autonomous systems (or discrete systems), red and green points are configured to be plotted alternately.
The following steps are used to stop the display of the above-mentioned green points.
- For an autonomous or non-autonomous systems, select Setting -> Poincare. For discrete systems, select Setting -> Map.
(a) For autonomous systems or non-autonomous systems.
(b) For discrete systems.
- Start up the Configuration window.
(a) For the autonomous system.
(b) For the non-autonomous system.
(c) For the discrete system.
- Change the submap switch to OFF in the Configuration window (remove check)*4
- The following shows the difference in the drawn figure when the submap switch is On and when the switch is Off.
(a) For the submap
On
(b) For the submap
Off
How to change the color of the points of the Poincare map (periodic points) in the drawing
For example, after starting from a point on the Poincare section, a red point is plotted the next time the orbit crosses the Poincare section. If the orbit of the limit cycle crosses the Poincare section multiple times, using the default settings, the points are plotted as green points except the point for the first time the orbit crosses the Poincare section. Therefore, when a Poincare section intersecting a limit cycle eith the period-2 is configured, red and green points are plotted alternately on the orbit drawn in the drawing window. Furthermore, for 2-periodic solutions (2-periodic points) in non-autonomous systems and discrete systems, red and &color(green){green} points are configured to be plotted alternately as well.
The following steps are used to change the color of points that would otherwise be plotted as green points.
- For autonomous and non-autonomous systems, select Setting -> Poincare. For discrete systems, select Setting -> Map.
- Start up the Configuration window.
- Press the color button in the submap in the Configuration window
- Start up the Color palette.
- Select the color to change to, or directly input RGB values. Press the OK button to finish configuration after selection.
- In the following example, the color is changed from green to blue.
(a) Default setting
(b) After the change of the point color