Henon_map_en

Henon map †

For example, let us consider the Henon map:

and calculate the saddle-node bifurcation set of a 5-periodic point in the -plane.

In the following, we make the following programs, and use these:

1. PP
2. FIX
3. BF

Configurations †

• First, please refer to the Installation section to install BunKi, then go to the Preferences.
  • After completing the Preferences settings continue to Creations of the Programs.
• Complement. 1
  • If the Symbolic Math Toolbox (MATLAB additional package) is not available, please consult the following sample for a demo. It is a file where we gathered all the analysis tools required for the analysis of the Henon map, and the configuration files in a compressed project directory. Please extract its content in any location.
    • Henon-sample.zip
  • Attention：starting BUNKI from the MATLAB command in the expanded directory, the SE program starts. If choosing "system create" there, an install without the "Symbolic Math Toolbox" will return an error. If the "Symbolic Math Toolbox" is not available, please skip the Creations of the Programs and directly continue to PP.
• Complement. 2
  • In the extracted directory of the BUNKI program, there is a sample directory.
    

    /BUNKI/Sample/Henon_map

    The various files placed down this directory are used in the following explanations. It is possible to save some time by using such files instead of filling up each configuration field manually. The files prepared are as follows:
    • Henon.sed : file containing the model equations : configuration file for use with SE
    • 5Pd_point.ppd : The example of a stable 5-periodic point (No.1) : configuration file for use with PP (No.1)
    • chaos.ppd : The example of a chaos attractor (No.2) : configuration file for use with PP (No.2)
    • 5Pd_point.fixd : For the detection of the saddle-node bifurcation : configuration file for use with FIX
    • 5Pd_point_SN.bfd : Setup file for the calculation of the saddle-node bifurcation of a 5-periodic point : configuration file for use with BF (No.1)
    • 5Pd_point_PD.bfd : Setup file for the calculation of the period-doubling bifurcation of a 5-periodic point : configuration file for use with BF (No.2)
      
      
      

Creations of the Programs †

The following steps are done in a sub-directory. Any name can be used for the sub-directory. First, the following is executed in the MATLAB command window:

>> BUNKI

After the execution, a window named Select BUNKI Project Directory will open.

Here the Project to be analyzed is created or selected. At this stage, no Project is generated. Therefore, input a Project name for identification in the Name field and click the OK button. In this case, since the project is the analysis of the Henon map, Henon is used as the project name.*1

When creating a new Project, you are asked whether it is okay to create a directory (or folder). Please select Yes. SE (system editor) will start.

We now create the programs for analysis.*2

• Description of system
  • First, we select what type of system is going to be analyzed. The Henon map has two variables and is an discrete system. Therefore, the Dimension is set to 2，and Discrete is checked as the system type.
    
    
    
    
    
  • The equations for the Henon map is input into the Equations field. Please follow the following rules.
    • State variables : Each state variable should be written as x[i], i = 1,2,...,n . In other words, equations are input using the substitutions x -> x[1]，y -> x[2].
    • Parameters : Parameters are placed between two $ symbols to identify that the variable is a parameter. For example, items placed between two $ symbols, such as $para$，$Vm$，$alpha$，etc, are considered parameters. Parameter names can be chosen freely.
      
      
      
      
      

• Syntax check & program creation
  • The above figure shows the screenshot after the input of the system equation. Click the check button to check whether there are no inconsistencies in the equations. The syntax checking will start.
    
    
    
    
    
  • After the syntax check, if there are no errors, then programs are created. The bottom-right check button controls which programs are created. To obtain a bifurcation curve, check all of PP，FIX，BF. Please click Create button to start program creations.
    
    
    
    
    (a): The status bar will be displayed when the system creation starts.
    
    
    
    
    (b): After system creation, a message is shown to indicate that the programs were created without problem in the log window.
    
    

PP †

Here, we import the initial values of parameters and state variables to draw a phase portrait and to capture a stable periodic point.

First, start the PP (phase portrait) program from SE. Select Program -> PP.

The drawing window and the control panel will start. In the first run, the drawing window is the (x[1],x[2]) plane.*3 Here, there is no need to change the drawing window.

Next, we deal with the control panel. First, we input the values of Initial states and Parameters as the initial condition.*4. The initial points to start the simulations are input (x[1],x[2]) in the Initial states field. Here, let us start from the origin (0,0) !

Please do not forget to input the parameter values before drawing. When first ran, 1 is entered for all parameters. Parameters are configured as follows.

Next, click the Start button to start simulations. Can you see that a solution starting from the origin converges to a periodic point?

Since transient responses are displayed in the drawing window, it is difficult to understand in the state. The point sequence during transient response can be erased by clicking the Clear button.

By erasing the point sequence for transient response, we see that the dynamics of the system convergence to a stable 5-periodic point. Therefore, let us set the period of the periodic point to 5 periods. The Period in Plot field in main panel is changed from 1 to 5. With this change, red points are displayed every time the Henon map repeats five maps. Other points are displayed as green points.*5

Parameters can be changed in real time. Let us change the parameter a. You can observe that by decreasing the parameter a, the stable 5-periodic point is changed to a chaotic state.

Next, we resume simulations after changing the parameter a to the initial value 1.435 . Check the check box the states flow in the Plot field. In this way, we can check the current values of the state variables in the list box. State variables for each map is displayed in the list box. In this example, we can see to converge to a stable 5-periodic point.

Next, let us investigate bifurcations of this 5-periodic point. The eigenvalues of the variational equation for the 5-periodic point are evaluated when the parameters are changed. For this purpose, the FIX tool can be used. We pass the initial values to use FIX from PP. The important thing is that the values of the state variables and parameters at the steady state must be passed to FIX.

To confirm whether the system reached a steady state, either check the check box states flow switch and confirm that the values of the state variables no longer change as described above, or click the Status button a few times and confirm that the information displayed in the list box does not change.

After confirming that there are no more changes, select Export current status from Tools in the menu bar. With this procedure, the values of state variables, the parameter values, and other configuration values at the steady state are preserved and can be passed to the next program (see section on FIX).

(a): Select export.

(b): The export of configuration values finished.

FIX †

Here, the objective is to track the accurate location of a periodic point when the parameters are changed and also to monitor the eigenvalues during tracking.

Users must adjust the configurations to run FIX. For simplicity, here let us run FIX as is. Normally, the FIX program is run after the values saved by the PP program (using Export Current status) is imported as the initial values for Newton’s method.*6

First, let us start up the FIX program from the PP control panel. Select Program -> FIX.

When initially starting up the FIX, default values are entered in each field.

Information on the periodic point obtained with PP is imported as the initial configuration value. Select Tools ->Import initial point.

After the selection, the information obtained with PP such as the parameter values and the coordinates of the periodic point are reflected.

By using PP, we saw that the dynamics of the system changes by decreasing the parameter a. Therefore, select the parameter to be changed as a and configure the step size to decrease the parameter.

Click the Start button to start calculations. A new window will open when the calculations start. This window is to monitor the location of the eigenvalues on a Gauss plane. The saddle-node bifurcation of a periodic point occurs when one of the eigenvalues becomes "unity". We can observe that as the parameter changes, the location of the eigenvalues change, and one eigenvalue on the window approaches "unity".

Parameters are continuously changed, and when a bifurcation point is detected, the program stops.*7 In this example, the saddle-node bifurcation is detected because one of the eigenvalues approached 1 and Newton’s method did not converge.

The values of the parameters and the state variables when the program stopped can be used as the initial point for BF, the program to track bifurcation curves. The information when the bifurcation point is detected are exported for BF. When Export BF point is selected here, the status, parameter values, and other necessary configuration values are preserved. The preserved values are passed to the next program (see section on BF).

(a)： Select the Export BF point.

(b)： The exported information is displayed in the list box.

BF †

The BF tool is used to obtain a bifurcation curve by changing system parameters.

The user must adjust the configuration to run BF. For simplicity, here let us run BF as is. Normally, the BF program is run after the values of the state variables and parameters at the bifurcation point detected by the FIX program are imported as the initial values for Newton's method.*8

First, let us run the BF program from the FIX main panel. Select Program -> BF.

When starting up BF at the first time, default values are entered in each field.

Various information on the bifurcation point obtained with FIX is imported as the initial value. Select Tools ->Import BF point from main panel.

After the selection, various information such as the type of bifurcation and the period of the periodic point are input automatically into each field. To tracke the bifurcation set on the (a,b) plane, check the radio button a as the X parameter and b as the Y parameter. Furthermore, configure items such as step size to change the parameters and the value to stop the BF when reaching end parameter. To save the calculated results as a data file, input the output file name in Output.

After these configurations, click Start button to start tracking of the bifurcation curve. Then, the Eigenvalues window and Bifurcation diagram window will open as new windows. The Eigenvalues window is used to visualize the change in the root locus of the eigenvalues during calculation of the bifurcation curve. Furthermore, to monitor the change of the root locus, we can visually understand occurrence of a bifurcation point with co-dimension 2. The Bifurcation diagram window is a window to monitor how the saddle-node bifurcation is being tracked in real time.

(a)：The root locus of eigenvalues and the status of the list box when BF runs.

(b)：Bifurcation diagram in the (a,b)-plane. The calculation result can be monitored in real time.