How to obtain initial values to use FIX

Some kind of attractor information is necessary to use the FIX tool to search bifurcation points. The attractor information is used as initial conditions for Newton's method calculations, and one objective to run PP is to obtain the approximate value of the attractor.

It is necessary to find attractors (equilibrium points/limit cycles”¤periodic solutions and map points (fixed points), etc) that we are going to find bifurcation properties for, and pass information such as parameters and state variables at that time to FIX.

Therefore, simulations on the system are necessary to obtain initial values to run FIX. The following sections can be referred to.

The important thing is whether the solution converged to an equilibrium point or fixed points /periodic points of the Poincare map. In other words, the state variables and parameters when a steady state is reached have to be passed to FIX. For methods able to confirm this, see How to reflect states when the attractor reached a steady state to each field.

How to check whether the attractor reached a steady state

There are two methods to check whether a steady state was reached:

• Using the States flow switch
• Using the Status button

These two methods are explained below.

• Using the States flow switch
• This method is used to visually check the current list of state variables by continuously displaying the generated numerical integrals and maps in a list box. For autonomous and non-autonomous systems, the state variables list during numerical integration is displayed, and for discrete systems, the state variables list during map generation is displayed.
1. Check the states flow in Plot field of the PP main panel.

2. Press the Start button and start simulation.

3. The current state variables will be displayed in the list box.

4. Confirm that the value of state variables no longer change.

• Using the Status button
• This method is used to visually check the current list of state variables by displaying the generated numerical integrals and maps in a list box through an event. For autonomous and non-autonomous systems, the list of the state variables after numerical integration is displayed, and for discrete systems, the list of the state variables after map generation is displayed.
1. Press the Start button and start simulation.

2. Click the Status button. The state variables will be displayed in the listbox when the Status Button is clicked.

3. Confirm that the value of state variables no longer change after clocking the Status Button a few times.

How to reflect states when the attractor reached a steady state to each field

During analysis, there would be requests to

• Skip transient responses while making data of a phase portrate, a time course, and so on.
• Restart simulation after convergence to an equilibrium point/a periodic solution.

Here, if PP is stopped once and the simulation is to be restarted, by default, simulation starts not from the discovered attractor, but from the conditions before finding the attractor (in other words, from the initial condition described in the Initial states field in the Main panel). Therefore, there would be requests to reflect the conditions when the attractor was discovered to the initial value and parameter fields. The following shows the steps for the reflection procedure.

Two methods are available to reflect the values of state variables and parameters to the respective fields when the attractor reached a steady state:

• Using the mouse,
• Using the Menu Bar: Tools.

Use preferred method*2.

• Method using mouse
1. Confirm that the state variables no longer change because some kind of the attractor reached an equilibrium point/a fixed point. (see How to check whether the attractor reached a steady state).

2. Right click in list box.*3 A selection menu will be displayed. Select Export current status.

3. Confirmation that the current statuses are stored will be displayed in the list box.

4. Right click in the list box again, and select Import status.

5. The current status will be reflected in each field (the values of state variables and parameters).

(a) Main panel after the reflection

(b) Initial states and parameters before the reflection

• Using the Menu bar:Tools
1. Confirm that the state variables value no longer change because the attractor reached an equilibrium point/a fixed point. (see How to check whether the attractor reached a steady state).

2. Select Tools -> Export current status.*4

(a) Select Export current status

(b) Confirmation that is temporary stored will be displayed in the list box.

3. Next, select Tools -> Import status.

4. The current status will be reflected in each field (the values of the state variables and parameters).

(a) Main panel after the reflection

(b) Initial states and parameters before the reflection

How to observe equilibrium points(Autonomous system)

Check whether the parameters are set values at which an equilibrium point can be ovserved. If it is set the parameter at which the existence of an equilibrium point is known, just press the Start button afterwards.

The following are the steps when there is no information whether equilibrium points or limit cycles exist.

1. Configure axis to draw figure. State variables are selected from Menu: Setting->window as X-axis and Y-axis by the method given in How to change the phase plane of the phase portrait to be drawn.

2. Configure to arbitrary parameter values and press the Start button.

3. If a stable equilibrium point exists, the orbit of a solution converges to the equilibrium point.

4. To find where stable equilibrium points exist other than the converged equilibrium point, click an arbitrary position on the drawing window. The orbit of a solution with the clicked coordinates as the initial value will be displayed.*5

5. To confirm convergence to the equilibrium point, press the clear button in the Main panel and erase transient response.

(a) Display including transient response.

(b) The transient response is erased. The position of the equilibrium point will be displayed with a black point.

How to observe limit cycles (Autonomous system)

Check whether configured parameters result in a limit cycle. If existence of a limit cycle is known, just press the Start button afterwards.

The following are the steps when there is no information whether limit cycles are generated.

1. Configure axes to draw figure. Select Menu:Setting->window, and Select the state variables as X-axis and Y-axis by the method given in How to change the phase plane of the phase portrait to be drawn.

2. Configure to appropriately values of system parameters and press the Start button.

3. If a stable limit cycle exists, the orbit of the solution that starts from an arbitrary initial condition converges to the limit cycle.

4. To find where attractors exist other than the converged limit cycle, click an arbitrary position in the drawing window. The orbit of a solution with the clicked coordinates as the initial value will be displayed.*6

5. To confirm convergence to the limit cycle, press the Clear button in the Main panel and erase the transient response.

(a) Display of the limit cycle with the transient responses.

(b) The transient response is erased. If the position of the point on the Poincare section is shown together with the orbit of the limit cycle, it will be shown by a red point.

How to observe periodic solutions(Non-autonomous system)

In a non-autonomous system with an external periodic force, an entrained periodic oscillation to the periodic external force may be observed, depending on the parameter choice. If the parameters that result in a periodic solution are known, check whether the parameters are configured as such. Just press the Start button afterwards.

The following are the steps when there is no information whether a periodic solution can be observed or not.

1. Configure axes to draw figure. Select Menu:Setting->window, and Select the state variables as X-axis and Y-axis by the method given in How to change the phase plane of the phase portrait to be drawn.

2. Configure to appropriately values of system parameters and press the Start button.

3. If a stable periodic solution exists, the orbit of a solution that starts from an arbitrary initial condition converges to the periodic solution.

4. To find where attractors exist other than the converged limit cycle, click an arbitrary position in the drawing window. The orbit of the solution with the clicked coordinates as the initial value will be displayed.*7

5. To confirm convergence to the periodic solution, press the Clear button in the Main panel and erase the transient response.

(a) Display of the limit cycle with the transient responses.

(b) The transient response is erased. If the position of the periodic points of the Poincare map is shown together with the orbit of the periodic solution, it will be shown by a red point.

How to observe time course of periodic points (fixed points)(Discrete system)

In a discrete system, a sequence of points obtained by the mapping is plotted on the phase portrait. The methodology is similar to the one described in How to observe periodic solutions.

The following will show the steps to display the time course of the periodic points observed in a discrete system using an example of the Henon map.

1. Configure axes to draw figure. From Menu, select Setting->window, and discrete time n as the x-axis and the state variable x[1] as the Y-axis by the method given in How to display time evolutions of solutions as a waveform

2. Configure to appropriately values of system parameters and press the Start button.

3. The time course of the mapping points will be displayed.

4. Simulate the dynamical behaviors of the discrete system by changing the values of parameters. The following example is the time courses when one of the system parameters is changed. When the parameter is inceased, a 2-periodic point is changed to a 4-periodic point due to an occurence of the period-doubling bifuration.

(a) The case that a parameter is increased.

(b) Example of the generation from a 2-periodic point to a 4-periodic one.

How to show / hide the points of the Poincare map

Analyses of limit cycles or periodic solutions can be reduced to the analysis of the fixed or periodic point of the Poincare map. Therefore, some pieces of information must be confirmed, such as where to set the Poincare section on the orbit of a solution, or which to be the point of the Poincare map on the configured section.

By default, when the Poincare section is configured, the point of the Poincare map are hidden at the first boot of time PP runs, right after the PP tool are produced by SE tool. Of course, in discrete systems, the default configuration is that the points are shown and the information of the orbit between point sequences are omitted.

The following gives the steps to show or hide the points of the Poincare map.

1. For autonomous systems, the Poincare section must be configured transversely for the limit cycle. Refer to How to configure the Poincare section for configuration details. For a non-autonomous system, the points of the Poincare map correspond to the sampling points of the states at the period , and therefore, no special configuration is necessary.
2. Check the checkbox Poincare in Plot field of Main panel.

3. Points will be shown on the orbit of a solution.

(a) When points of the Poincare map are hidden.*8

(b) When the points of the Poincare map are shown. The red points are represented the points of the Poincare map.

4. To hide the points of the Poincare map, just uncheck the checkbox Poincare. This procedure can be done while a simulation is running.

How to confirm the period of limit cycles

How long the period of a limit cycle is becomes important when considering various problems, for, instance, the period of a repetitive firing in a neuron model, or the period of the circadian rhythm. The following steps can be used to check the period of the limit cycle currently observed.

1. Observe the limit cycle together with the points of the Poincare map. It is necessary to predetermine where to set the Poincare section on the limit cycle. For configurations of the Poincare section, see How to configure the Poincare section.

2. Confirm that the system reached a steady state.

(a) Convergence to a limit cycle.

(b) Confirm that the values of the state variables shown in the list box are constant values.

3. Press the Status button. The current status will be displayed in the list box of the Main panel. Then the information of the period of the limit cycle (Return time) will be displayed as additonal information.*9

How to confirm the period of periodic solutions

How long the period of a periodic solution (for non-autonomous systems) or a periodic point (for discrete systems) is becomes important when detecting bifurcation points and calculating bifurcation sets. The following steps can be used to check the period of the solution currently observed.

1. Start simulation with the point of the Poincare map shown.

2. In this example, a 4-periodic point is displayed.

3. When multiple-periodic points are displayed, check Period in the Plot field of the Main panel. Default configuration is "£±".

4. Period 1 indicates a red point will be displayed per one map. We can easily see that this is a 4-periodic point, and therefore the Period is changed to "4".

5. With this change, one red point will be displayed per four maps. The points of the second, third, and fourth map will be displayed with green points.

Here, if the period of a periodic point is difficult to determine visually, it is necessary to repeatedly increase the value of Period and determine the period by trial and error.*10 If the correct period is configured, only a unique red fixed point will be displayed.

1. When the period is set to "2", one red point will be shown for every two maps. When the period is set to "3", the red points will be seen moving around.

(a) Display when the Period is set to "2".

2. When the correct period is configured, only one red point would be displayed. Therefore, the value of the Period will be increased.
3. The accuracy can be increased by showing points of the map once per every configured period in the display window. Refer to How to control the number of the points of the Poincare map (periodic points).

(a) Turn off the display of green points.

(b) The point of the map will be shown only once every four periods.

*1 Here, and throughout this web site, the periodic oscillation observed in an autonomous system is called limit cycle, while in a non-autonomous system, that is called periodic solution.
*2 It's also able to input manually the displayed values in the list box into the Initial state field. However, if the number of the state variables is increased, it hards enough to do so.
*3 In this tiem, it recommends to click Pause Button. On the other hand, if the Stop Button is clicked, then the calcuration is stopped. So note that the correct values are not stored.
*4 In this tiem, it recommends to click Pause Button. On the other hand, if the Stop Button is clicked, then the calcuration is stopped. So note that the correct values are not stored.
*5 The PP tool can be also simulated the behaviors of the system by changing the values of the Initial states feild in the main panel.
*6 The PP tool can be also simulated the behaviors of the system by changing the values of the Initial states feild in the main panel.
*7 The PP tool can be also simulated the behaviors of the system by changing the values of the Initial states feild in the main panel
*8 A black point will be shown. This is not shwon the point of the Poincare map. Note that it is displayed the position of the state variables when the simulation is started at and/or the Clear button is clicked.
*9 In fact, to set the state flow switch and to display the values of the state variables in the list box, the period of the limit cycle is also shown. The values shown in the list box indicate time, state variables x[1], x[2], ... , x[n], color(magenta){period}; in order.
*10 An automatical decision function will be implemented in the future.