Initial Conditions

To solve boundary-value problems using shooting, click here

(R)ange
(2)par range
(L)ast
(O)ld
(G)o
(M)ouse
(S)hift
(N)ew
s(H)oot
(F)ile
form(U)la
m(I)ce
(D)AE guess
(B)ackward

• (R)ange
  This lets you integrate multiple times with the results shown in the graphics window. Pressing this option produces a new window with several boxes to fill in.

  First choose the quantity you want to range over. It can be a parameter or a variable. The integrator will be called and this quantity will be changed at the beginning of each integration.

  Then choose the starting and ending value and the number of steps.

  The option Reset storage only stores the last integration. If you choose not to reset, each integration is appended to storage. Most likely, storage will be exceeded and the integration will overwrite or stop.

  The option to use last initial conditions will automatically use the final result of the previous integration as initial data for the next integration. Otherwise, the current ICs will be used at each step (except of course for the variable through wich you are ranging.)

  If you choose Yes in the Movie item, then after each integration, XPP will take a snapshot of the picture. You can then replay this series of snapshots back using the Kinescope.

  When you are happy with the parameters, simply press the OK button. Otherwise, press the Cancel button to abort.

  Assuming that you have accepted, the program will compute the trajectories and plot them storing none of them or all of them. If you press Esc it will abort the current trajectory and move on to the next.

  Pressing the / key will abort the whole process.

• (2)par range
  Similar to range integration, but allows you to range over two items.

  The Crv(1) Array(2) item determines how the range is done. If you choose Crv then the two paramaters are varied in concert, [a(i),b(i)] for i=0,...,N.

  The more useful Array varies them independently as [a(i),b(j)] for i=0,...,N and j=0,...,M.

• (L)ast
  This uses the end result of the most recent integration as the starting point of the current integration.

• (O)ld
  This uses the most recent initial data as the current initial data. It is essentially the same as Go.

• (G)o
  Uses the initial data in the IC window and the current numerics parameters to solve the equation.

  The output is drawn in the current selected graphics window and the data are saved for later use.

  The solution continues until either the user aborts by pressing Esc , the integration is complete, or storage runs out.

• (M)ouse
  allows you to specify the values with the mouse. Click at the desired spot in a phase-plane picture. You must have a two-dimensional view and only variables along the axes.

• (S)hift
  This is like Last except that the stating time is shifted to the current integration time. This is irrelevant for autonomous systems but is useful for nonautonomous ODEs.

• (N)ew
  This prompts you at the command line for each initial condition.

  Press Enter to accept the value presented. Press Esc to quit entering and start the integration.

• s(H)oot
  allows you to use initial data that was produced when you last searched for an equilibrium.

  When a rest state has a single positive or negative eigenvalue, then XPP will ask if you want to approximate the invariant manifold. If you choose yes to this, then the initial data that were used to compute the trajectories are remembered. Thus, when you choose this option, you will be asked for a number 1-4. This number is the order in which the invariant trajectories were computed.

  1 and 2 are always unstable manifolds and 3 and 4 are stable manifolds.

  Note if the invariant set is a stable manifold, then you should integrate backwards in time.

• (F)ile
  Prompts you for a file name which has the initial data as a sequence of numerical values.

• form(U)la
  Allows you to set all the initial data as a formula. This is good for systems that represent chains of many ODEs.

  When prompted for the variable, type in u[2..10] for example to set the variables u2,u3, ..., u10 and then put in a formula using the index [j] .

  Note you must use [j] and not j by itself.

  For example sin([j]*2*pi/10). Repeat this for different variables hitting enter twice to begin the integration.

• m(I)ce
  allows you to choose multiple points with the mouse. Click Esc when done.

• (D)AE guess
  lets you choose a guess for the algebraic variables of the DAE.

• (B)ackward
  is the same as Go but the integration is run backwards in time.