• Solver : Selection and detailed settings of numerical integration.


    • Solver : selection of ODE solver built in MATLAB is possible.


      ode45This is based on an explicit Runge-Kutta formula, the Dormand-Prince pair. It is a one-step solver; that is, in computing y(tn), it needs only the solution at the immediately preceding time point, y(tn-1). In general, ode45 is the best solver to apply as a "first try" for most problems.
      ode23This is also based on an explicit Runge-Kutta proposed by Bogacki and Shampine. It may be more efficient than ode45 at crude tolerances and in the presence of mild stiffness. ode23 is a one-step solver.
      ode113This is a variable order Adams-Bashforth-Moulton PECE solver. It may be more efficient than ode45 at stringent tolerances. ode113 is a multi-step solver; that is, it normally needs the solutions at several preceding time points to compute the current solution.
      ode23s};This is based on a modified Rosenbrock formula of order 2. Because it is a one-step solver, it may be more efficient than ode15s at crude tolerances. It can solve some kinds of stiff problems for which ode15s is not effective.
      ode15sThis is a variable order solver based on the numerical differentiation formulas (NDFs). These are related to but are more efficient than the backward differentiation formulas, BDFs (also known as Gear's method). Like ode113, ode15s is a multi-step method solver. If you suspect that a problem is stiff or if ode45 failed or was very inefficient, try ode15s.
      myselfThis is a standard fixed step 4th order Runge Kutta method.It is not recommended unless under particular circumstances.

  • Relative tolerance : measures the error relative to the size of each state. The relative tolerance represents a percentage of the state's value.
  • Absolute tolerance : s a threshold error value. This tolerance represents the acceptable error as the value of the measured state approaches zero*1.
  • Step type : selection of the step size method used by the Solver
    • Variable : use a variable step size (Default)
    • Invariable : use a fixed step size
  • Initial step : By default, the solvers select an initial step size by examining the derivatives of the states at the start time. If the first step size is too large, the solver may step over important behavior. The initial step size parameter is a suggested first step size. The solver tries this step size but reduces it if error criteria are not satisfied.
  • Solver frequency : Time for the solver to complete one calculation. After one calculation, the solver will interpret the input and keyboard events only after the time defined by the frequency, and carry on. Key response, click response are related to how much we display at once. That is to say, if we set this to a small value, the amount of generated events will increase, and on the contrary, a bigger value will decrease it. In the case of an extremely slowly responsive system's PP, increasing the frequency (decrease this value) will not have any effect, while in the case of a fast system, you will need to make this value smaller if you want to input events in the middle of an attractor.
  • Time direction: The time direction to run the numerical integration. + is the positive direction of the time, while - is the negative drection of one.


  • Parameter : associate a key to a particular parameter adjustment.

    • You can assign keys to parameters in the alphabetic order.You can also select a key for each parameter in the associated pull tab.
    • a parameter increase is associated to an upper case letter,a decrease to a lower case letter.
    • the default value for both increase or decrease granularity is set to 0.1. In case of modification, please enter the desired value.


  • Window : drawing window's setting options


    • Add New Window : in case you want to create an extra drawing window, clicking this button will create a new one.


    • X-axis : you can select as the x-axis either time t or any of the state variables, x[1], x[2], ... , x[n].
    • Y-axis : you can select as the y-axis any of the state variables, x[1], x[2], ... , x[n].
      • range : the default setting Auto-scale of the drawing range adjusts this setting automatically. However, if you want to visualize the system response in a fixed range, this option makes it possible.


    • Line : Adjustment of the drawing line width and color.
      • line color : selection of the line color. It can be set through the RGB value.It is also possible to press the line color button to display a color palette.You can then select any color from this palette.
      • width : setting of the line width. The unit is pt.
    • Point : adjustment of the size and color of the points representing potential fixed points or periodic points.
      • point color : selection of the point color. It can be set through the RGB value.It is also possible to press the point color button to display a color palette.You can then select any color from this palette.
      • size : setting of the point size. The unit is pt.


  • Poincare : detailed settings of the Poincare section.


    • Poincare section method : depending on the characteristics of the problem, 3 different methods can be selected.


      • Fixed Value : For a state variable, inputs an arbitrary value at which the limit cycle crosses transversely. From the pull down menu, select the state variable'' as the Poincare section. Then enter the set value for the Poincare section. The default value is x[1] = 0.

      • Equilibrium : Limit cycle generates from an equilibrium point which became unstable.Therefore, a limit cycle can be expected to be in the neighborhood of the unstable equilibrium point. Then, by composing a Poincare section based on the unstable equilibrium point, it is possible to avoid the scenario where the orbit of the limit cycle comes into contact with the defined Poincare section. The method of the setting is as follows:
      1. By pressing the search button, a panel to input the initial values for searching an unstable equilibrium point starts up.


      2. To give appropriately initial values and press the calculate button, it starts to search an equilibrium point. If you give the appropriately initial values, you will get the coordinate of the equilibrium point.


      3. Based on the obtained coordinate of the equilibrium point, the Poincare setion is set. Select the state variable as the Poincare section.


      • Equation*2 : In case the limit cycle possesses some symmetries in the phase space, defining the Poincare section through a given function can be beneficial under certain circumstances. This mode is dedicated to such circumstances.


    • direction : settings for the direction when the Poincare sections and the orbit intersect.
      • + : display the point when the orbit crosses the Poincare section from the top.
      • - : display the point when the orbit crosses the Poincare section from the bottom.
      • +&- : display the both point when the orbit crosses the Poincare section at the top and the bottom.


    • return time max : Maximum value of the Return Time. This can be used to determine if the observed attractor is an equilibrium point or a limit cycle. If the Return time is longer than the chosen maximum, it is likely the orbit converged toward an equilibrium, thus the observed attractor can be considered an equilibrium point. Therefore, there are some situations where the period of limit cycles is particularly long, requiring a large maximum return time value.


  • sub map: This option is used to control the display of the point of the Poincare map. The limit cycle that starts from the point on the Poincare section will hit the Poincare section again. Then, the point is displayed as a red point. In the case that the limit cycle has multi-cross point, other points than the first point is represented as green point in the default setting. The sub map switch decides whether or not represents the green point(Default setting is On.).


    • color : selection of the point color of the other point than the first crossing point. It can be set through the RGB value.It is also possible to press the point color button to display a color palette.You can then select any color from this palette.


  • Equation : this option is used to verify the equations of the analyzed autonomous system.


*1 for detail see odeset
*2 This is a special mode. It should not be used unless the user is sure it is necessary.

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Last-modified: 2009-07-23 (Thu) 20:13:30 (3767d)