```LEFT:[[aboutPP>aboutPP_en]]
CENTER:[[PP>PP_en]]
RIGHT:[[User's Guide>users_guide_en]]~
&br;&br;

&aname(PP_top);
&br;&br;

-&size(20){[[How to obtain initial values to use FIX>aboutPP_operations_en#pptofixexport]]};
-&size(20){[[How to reflect states when the attractor reached a steady state to each field>aboutPP_operations_en#pp_import]]};
-&size(20){[[How to observe equilibrium points>aboutPP_operations_en#pp_show_eq]](Autonomous system)};
-&size(20){[[How to observe limit cycles>aboutPP_operations_en#pp_show_lc]]((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''.))(Autonomous system)};
-&size(20){[[How to observe periodic solutions>aboutPP_operations_en#pp_show_period]](Non-autonomous system)};
-&size(20){[[How to observe time sequence of periodic points (fixed points)>aboutPP_operations_en#pp_show_fix]](Discrete system)};
-&size(20){[[How to show / hide the points of Poincare map>aboutPP_operations_en#pp_show_poincare]]};
-&size(20){[[How to confirm the period of limit cycles>aboutPP_operations_en#pp_check_tau]]};
-&size(20){[[How to confirm the period of periodic solutions (periodic point)>aboutPP_operations_en#pp_check_period]]};~
&br;&br;

&aname(pptofixexport);
LEFT:&size(20){''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>PP_en]] is to obtain the approximate value of the attractor.
//FIX ツールを動かす，すわなち分岐点の探索を行うためには，なんらかのアトラクタ情報が必要となる．それは Newton 法の初期値として利用するための情報であり，そのための近似値を得ることが [[PP>PP]] を実行する一つの目的でもある．

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.
//これから分岐を調べようと思っているアトラクタ(平衡点・リミットサイクル，周期解，写像点(固定点)など)を探し出し，その時のパラメータ値，状態変数値といった情報を FIX へ渡す必要がある．

Therefore, simulations on the system are necessary to obtain initial values to run FIX. The following sections can be referred to.
-For an autonomous system，[[How to observe equilibrium points>aboutPP_operations_en#pp_show_eq]],or [[How to observe limit cycles>aboutPP_operations_en#pp_show_lc]]
-For a non-autonomous system，[[How to observe periodic solutions>aboutPP_operations_en#pp_show_period]]
-For a discrete system, [[How to observe the time course of periodic points (fixed points)>aboutPP_operations_en#pp_show_fix]]
//従って，FIX を動かすための初期値を得るには，
//などを参考に系のシュミレーションを行う必要がある．

The important thing is whether the solution converged to an equilibrium point or fixed points /periodic points of the Poincare map. In other words, &color(magenta){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>aboutPP_file_en#pp_import]].

&br;&br;
&br;&br;

LEFT:&size(20){''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.~
&br;&br;

-&size(15){''Using the States flow switch''};~
--This method is used to visually check the current list of state variables by &color(magenta){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.
//--この方法は，数値積分，写像の生成状態をリストボックス内に&color(magenta){連続的に表示};することで，現在の状態変数値リストを視覚的にチェックする方法である．自律系・非自律系方程式であれば，数値積分を行った際の状態変数リストを，離散系方程式であれば，写像生成時の状態変数リストを表示する．

+++ Check the ''states flow'' in Plot field of the PP main panel.~
&br;
&br;
+++ Press the ''Start button'' and start simulation.~
&br;
&br;
+++ The current state variables will be displayed in the list box.~
&br;
&br;
+++ Confirm that the value of state variables no longer change.~
&br;
&br;

-&size(15){''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.
//--この方法は，&color(magenta){イベントを介して};，数値積分，写像の生成状態をリストボックス内に表示することで，現在の状態変数値リストを視覚的にチェックする方法である．自律系・非自律系方程式であれば，数値積分を行った際の状態変数リストを，離散系方程式であれば，写像生成時の状態変数リストを表示する．

+++ Press the ''Start button'' and start simulation.~
&br;
&br;
+++ Click the ''Status button''. The state variables will be displayed in the listbox when the Status Button is clicked.~
&br;
&br;
+++ Confirm that the value of state variables no longer change after clocking the Status Button a few times.~
&br;

&br;&br;
&br;&br;

&aname(pp_import);
LEFT:&size(20){''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.
//解析を進めていく上で，
//-データを作成する時に過渡応答部分を省略したい．
//-平衡点・周期解へと収束した状態から再度シミュレーションを開始したい．
//といった要求が出てくる．

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.
//さて，一度 PP を stop し，再度シミュレーションを開始する場合，何もしなければ，見つかったアトラクタからではなく，アトラクタを見つける前の状態(すなわちメインパネル内の Initial states 欄に記載された初期状態)から再度シミュレーションがスタートすることになる．そのため，アトラクタを見つけた時の状態を初期値・パラメータ欄にそれぞれ反映させておきたいといった要求が出てくる．以下では，その反映させるための手順を示す．

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'',

Use preferred method((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.)).
//定常状態へ至った時点での状態変数値，パラメータ値を各フィールドに反映させるには，
//-''マウスを利用''
//-''メニューバー：''Tools'' を利用''
//の２通りの方法がある．好みの方法を用いれば良い((リストボックス内に表示された数値を状態変数欄に手動で書き込むことももちろん可能ではあるが，変数の数が増えるとこれは大変である．))．

&aname(pp_import1);
-&size(15){''Method using mouse''};~

++ 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>aboutPP_operations_en#pp_steadystate]]).~
&br;
&br;
++ Right click in list box.((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.)) A selection menu will be displayed. Select ''Export current status''.~
//リストボックス内で右クリック((この時，''Pause Button''を押してから実行するとよい．一方，''Stop Button'' を押してしまうと計算がストップしてしまい，正常な数値がストックされないので注意を要する．))．選択メニューが表示される．''Export current status'' を選択．~
&br;
&br;
++ Confirmation that the current statuses are stored will be displayed in the list box.~
//現在の状態がストックされたことがリストボックス内に表示される．~
&br;
&br;
++ Right click in the list box again, and select ''Import status''.~
//再びリストボックス内で右クリックし，''Import status'' を選択する．~
&br;
&br;
++ The current status will be reflected in each field (the values of state variables and parameters).~
//現在の status が各項目(状態変数値，パラメータ値)に反映される．~
&br;
&br;
(a) Main panel after the reflection~
//反映後のメインパネルの様子~
&br;
&br;
(b) Initial states and parameters before the reflection~
//反映前の Initial states と parameter~

&aname(pp_import2);

++ 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>aboutPP_operations_en#pp_steadystate]]).~
&br;
&br;
++ Select ''Tools'' -> ''Export current status''.((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.))~
&br;
&br;
(a) Select ''Export current status''~
&br;
&br;
(b) Confirmation that is temporary stored will be displayed in the list box.~
//リストボックス内に一時状態がストックされたことが表示される．~
&br;
++ Next, select ''Tools'' -> ''Import status''.~
&br;
&br;
++ The current status will be reflected in each field (the values of the state variables and parameters).~
//現在の status が各項目(状態変数値，パラメータ値)に反映される．~
&br;
&br;
(a) Main panel after the reflection~
&br;
&br;
//(b) 反映前の Initial states と parameter~
(b) Initial states and parameters before the reflection~

&br;&br;
&br;&br;

&aname(pp_show_eq);
LEFT:&size(20){''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.
//平衡点がみられるパラメータに設定がしてあるかどうかをチェック．もし発生することが判ってるのであれば，後はスタートボタンを押せばよい．

The following are the steps when there is no information whether equilibrium points or limit cycles exist.
//以下は，平衡点が発生するのか，リミットサイクルが発生するかどうかは全く未知である場合の方法である．

+ 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>aboutPP_window_en#pp_change_win]].~
&br;
&br;
+ Configure to arbitrary parameter values and press the Start button.~
//適当なパラメータに設定し, start ボタンを押す．~
&br;
&br;
+ If a stable equilibrium point exists, the orbit of a solution converges to the equilibrium point.~
//安定平衡点が存在すれば，解軌道は平衡点に収束する．~
&br;
&br;
+ 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.((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.))~
//収束した平衡点以外の安定平衡点が無いか？を探したければ，描画 Window 上の任意の位置をマウスでクリックする．これでクリックされた座標を初期値とした解軌道が描かれる((もしくはメインパネル内の状態変数値欄の数値を変更することで様々な場合のシミュレーションが行える．))．~
&br;
&br;
+ To confirm convergence to the equilibrium point, press the clear button in the Main panel and erase transient response.~
//平衡点へ収束した状態を確認するためには，メインパネル内 clear ボタンを押すことで過渡応答部分を消してくれる．~
&br;
&br;
(a) Display including transient response.~
&br;
&br;
(b) The transient response is erased. The position of the equilibrium point will be displayed with a ''black point''.
//過渡応答部分を消去．平衡点の位置は，''黒点''によって示される．

&br;&br;
&br;&br;

&aname(pp_show_lc);
LEFT:&size(20){''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.
//リミットサイクルがみられるパラメータ設定となってるかどうかをチェック． もし発生することが判ってるのであれば，後はスタートボタンを押せばよい．

The following are the steps when there is no information whether limit cycles are generated.
//以下は，リミットサイクルが発生するかどうかは全く未知である場合の方法である．

+ 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>aboutPP_window_en#pp_change_win]].~
&br;
&br;
+ Configure to appropriately values of system parameters and press the Start button.~
//適当なパラメータに設定し, start ボタンを押す．~
&br;
&br;
+ If a stable limit cycle exists, the orbit of the solution that starts from an arbitrary initial condition converges to the limit cycle.~
//安定なリミットサイクルが存在すれば，解軌道はリミットサイクルに収束する．~
&br;
&br;
+ 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.((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.))~
//収束したリミットサイクル以外のアトラクタが無いか？を探したければ，描画 Window 上の任意の位置をマウスでクリックする．これでクリックされた座標を初期値とした解軌道が描かれる((もしくはメインパネル内の状態変数値欄の数値を変更することで様々な場合のシミュレーションが行える．))．~
&br;
&br;
+ To confirm convergence to the limit cycle, press the ''Clear'' button in the Main panel and erase the transient response.~
//リミットサイクルへ収束した状態を確認するためには，メインパネル内 clear ボタンを押すことで過渡応答部分を消してくれる．~
&br;
&br;
(a) Display of the limit cycle with the transient responses.~
&br;
&br;
(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 &color(red){''red''}; point.
//過渡応答部分を消去．もし Poincare 写像点も併せて表示した場合は，''赤点''によって示される．

&br;&br;
&br;&br;

&aname(pp_show_period);
LEFT:&size(20){''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.
//周期的外力の印加された非自律系方程式には，パラメータ設定によって周期的外力に''同調''した周期解が観測できる．周期解がみられるパラメータ設定が判っている場合，そのパラメータ値になっているかどうかをチェックする． 後はスタートボタンを押せばよい．

The following are the steps when there is no information whether a periodic solution can be observed or not.
//以下は，周期解が発生するかどうかは全く未知である場合の方法である．

+ 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>aboutPP_window_en#pp_change_win]].~
&br;
&br;
+ Configure to appropriately values of system parameters and press the Start button.~
//適当なパラメータに設定し, start ボタンを押す．~
&br;
&br;
+ If a stable periodic solution exists, the orbit of a solution that starts from an arbitrary initial condition converges to the periodic solution.~
//安定な周期解が存在すれば，解軌道は周期解に収束する．~
&br;
&br;
+ 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.((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))~
//収束した周期解以外のアトラクタが無いか？を探したければ，描画 Window 上の任意の位置をマウスでクリックする．これでクリックされた座標を初期値とした解軌道が描かれる((もしくはメインパネル内の状態変数値欄の数値を変更することで様々な場合のシミュレーションが行える．))．~
&br;
&br;
+ To confirm convergence to the periodic solution, press the ''Clear'' button in the Main panel and erase the transient response.~
//周期解へ収束した状態を確認するためには，メインパネル内 clear ボタンを押すことで過渡応答部分を消してくれる．~
&br;
&br;
(a) Display of the limit cycle with the transient responses.~
&br;
&br;
(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 &color(red){''red''}; point.

&br;&br;
&br;&br;

&aname(pp_show_fix);
LEFT:&size(20){''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>aboutPP_operations_en#pp_show_period]].

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.
//以下では，Henon 写像を例にとって，差分方程式の場合の時系列データを表示する方法を示す．

+ 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>aboutPP_window_en#pp_show_wave]]
//描画する座標軸を設定する．このためには，[[時間波形を観るためには？>aboutPP_window#pp_show_wave]]に示した方法で，メニュー：setting -> window より x-axis に''離散時間'' n を選択，y-axis に状態 x[1] を選択する．~
&br;
&br;
+ Configure to appropriately values of system parameters and press the Start button.~
//適当なパラメータに設定し, start ボタンを押す．~
&br;
&br;
+ The time course of the mapping points will be displayed.~
//時系列が表示される． ~
&br;
&br;
+ 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.~
//興味のあるパラメータを変化させて系のシミュレーションを行う．以下の例は，パラメータ a を増加させたときに発生する２周期点，４周期点の時系列データを示している.~
&br;
&br;
(a) The case that a parameter is increased.~
&br;
&br;
(b) Example of the generation from a 2-periodic point to a 4-periodic one.~

&br;&br;
&br;&br;

&aname(pp_show_poincare);
LEFT:&size(20){''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.
//リミットサイクル，もしくは周期解の分岐を調べるには，Poincare 断面上の Poincare 写像点の解析に帰着させて行う．それゆえに，解軌道上の何処に Poincare 断面を設定するか？，設定された Poincare 断面上の写像点が軌道のどの位置にあるか？といった情報を確認する必要がある．

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.
//SE よりツールを生成した直後の PP 初回起動時には，Poincare 断面を設定した場合の写像点((非自律系の場合はストロボ写像点(詳しくは[[周期的非自律系の Poincare 写像>PoincareMap#non-autonomous]]を参照) ))を&color(red){''表示しない''};ように設定されている．もちろん、差分方程式系の場合は，写像点を表示し、点列間の Orbit 情報を省略するように設定されている．

The following gives the steps to show or hide the points of the Poincare map.
//その Poincare 写像点を表示もしくは非表示にする方法を以下に示す．
+ For autonomous systems, the Poincare section must be configured transversely for the limit cycle. Refer to [[How to configure the Poincare section>aboutPP_poincare_en#pp_set_poincare]] 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 &ref(eq_img01.gif);, and therefore, no special configuration is necessary.~
+ For autonomous systems, the Poincare section must be configured transversely for the limit cycle. Refer to [[How to configure the Poincare section>aboutPP_poincare_en#pp_set_poincare]] 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 &ref(http://bunki.sat.iis.u-tokyo.ac.jp:50080/BUNKI/index.php?plugin=attach&refer=aboutPP_operations&openfile=eq_img01.gif);, and therefore, no special configuration is necessary.~
+ Check the checkbox ''Poincare'' in Plot field of Main panel.~
//メインパネルの Plot 内チェックボックス ''Poincare'' にチェックを入れる．~
&br;
&br;
+ ''Points'' will be shown on the orbit of a solution.~
//解軌道上に''点''が打たれるようになる．~
&br;
&br;
(a) When points of the Poincare map are hidden.((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.))~
&br;
&br;
(b) When the points of the Poincare map are shown. The &color(red){red}; points are represented the points of the Poincare map.~
&br;
+ To hide the points of the Poincare map, just uncheck the checkbox ''Poincare''. This procedure can be done while a simulation is running.
//もちろん，写像点を''非表示とする''場合は，チェックボックス ''Poincare'' のチェックを外せばよい．この操作は，シミュレーションを実行している間でも操作可能である．

&br;&br;
&br;&br;

&aname(pp_check_tau);
LEFT:&size(20){''How to confirm the period of limit cycles''};
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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.
//リミットサイクルの周期がどの程度あるか？は様々な問題を考える上で重要になる(例えば，ニューロンモデルの繰り返し発火周期(発火周波数)や，サーカデアンリズムのリズム周期など)．そのため，現在観測されているリミットサイクルの周期がどれぐらいあるか？をチェックするためには以下の様にすればよい．

+ 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>aboutPP_poincare_en#pp_set_poincare]].~
//Poincare 写像点を表示するようにしてリミットサイクルを観察．もちろんリミットサイクルの何処に Poincare 断面を設定するかは予め決めておく必要がある．Poincare 断面の設定に関しては，[[Poincare 断面を設定するには？>aboutPP_poincare#pp_set_poincare]]を参照．~
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+ Confirm that the system reached a steady state.~
//定常状態に落ち着いたことを確かめる．~
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(a) Convergence to a limit cycle.~
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(b) Confirm that the values of the state variables shown in the list box are constant values.~
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+ 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 (&color(red){Return time};) will be displayed as additonal information.((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.))~
//''Status ボタン'' を押す．メインパネルのリストボックスに現在の状態が表示される．このとき，その周期情報(&color(red){Return time};)も併せて表示される((実は，states flow switch をオンにして，状態変数値をリストボックスに表示することで，リミットサイクルの周期も併せて表示される．リストボックス内に表示される数値は，時間，状態変数 x[1], x[2], ... x[n], &color(magenta){周期};の順で表示されている．))．~
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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.
//現在観測されている周期解(非自律系)や周期点(離散系)の周期が何周期であるか？は分岐点の検出，分岐集合の計算の際に重要になる．現在観測している解の周期をチェックするためには以下の様にすればよい．

+ Start simulation with the point of the Poincare map shown.~
//Poincare 写像点を表示するようにしてシミュレーションを開始．~
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+ In this example, a 4-periodic point is displayed.~
//この例では，４周期点が表示される．~
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+ When multiple-periodic points are displayed, check &color(red){''Period''}; in the Plot field of the Main panel. Default configuration is ''"１"''.~
//多周期点が表示されている場合，メインパネル内 ''Plot'' 欄にある &color(red){Period}; をチェックする．デフォルトの設定では，''"１"'' に設定されている．~
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+ Period 1 indicates a &color(red){''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"''.~
//このとき，Period 1 は，１回の写像の度に，その座標に&color(red){''赤点''};が打たれることを意味する．４周期点であることが容易に判るので，この Period を ''"4"'' に変更する．~
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+ With this change, one &color(red){''red point''}; will be displayed per four maps. The points of the second, third, and fourth map will be displayed with &color(green){''green points''};.~
//この変更によって，写像４回毎に&color(red){''赤点''};を打つようになる．２回目，３回目，４回目の写像点は，&color(green){''緑点''};で表示されるようになる．~
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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.((An automatical decision function will be implemented in the future.)) If the correct period is configured, only a unique red fixed point will be displayed.
//一方，周期点が何周期であるか？を目視により判断するのが困難な場合には，''Period'' の数値を１つづ増加させ，その解が何周期点であるかを試行錯誤的に調べる必要がある((周期の自動判別については，今後実装する予定．))．正確な周期が設定されれば，赤点で表示される点が，ある写像点に固定される．
+ 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.~
//Period を ''"２"'' に設定した場合，２回の写像毎に赤点を表示する．Period を ''"３"'' に設定した場合は，赤点の位置が次々に入替わる様子をみることができる．~
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(a) Display when the Period is set to ''"2"''.~
//Period を ''"２"'' に設定した場合の様子．~
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+ When the correct period is configured, only one red point would be displayed. Therefore, the value of the Period will be increased.~
//正確な周期が設定されれば，表示される赤点は１点のみ表示されるはずなので，更に Period の数値を上げる．
+ 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)>aboutPP_poincare_en#pp_poincare_control]].~
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(a) Turn off the display of green points.~
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(b) The point of the map will be shown only once every four periods.~

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