Bifurcation Diagram Plotter (Version 1.0)

Introduction

• Widening your browser for better view of both painting areas!
• Click and drag a rectangle in the either diagrams to zoom in the graph. But to get a better view of graph, change the Minimum and Maximum value of x for function, see the last tip.
• The Function Graph responds very fast, so put whatever options for all entries, but the function must be in right format.
• The Bifurcation Diagram responds slower. With the default parameter, it takes about 5 seconds for repainting in a Pentium 133 PC. But considering this is a much more comlicated graph (in fact, it is the mechanism behind the graph more comlicated), such waiting time is still durable ?
• Changing the Dimension or initial point will generally do not affect the responding speed, but they do affect the accuracy of the graph.
• Changing the Number of Points parameter will affect the responding time. More time for more points, but the time increasing is linear, so if you think it is worthing that you wait 20 seconds for 500 points, then try it !
• The worst thing is to change the Numerical mesh of computation parameter, say, from 0.01 to 0.001 or even 0.0001. Though it will result in much better accuracy of the graph, but it may take minutes of waiting, even the machine will stuck. So for accuracy of data, we have an alternate way.
• The Bifurcation Diagram will only plot the points whose y value is on the x value of Function Graph, so choose the Minimum and Maximum value of x for function, and get desired diagram.

Background Knowledge of this Java Applet

where \lambda is a positive parameter. This boundary value problem is reduced from a semilinear elliptic equation:

(2)
           \Delta u+\lambda f(u)=0,    in \Omega,
            u=0,                                  on \partial \Omega,

where \Omega is  the unit ball in Rⁿ  .  By a remarkable paper by Gidas, Ni and Nirenberg [GNN], the solution of equation (2)
is always radially symmetric, if f(u) is local lipshitz in R⁺  . Thus the solution to (2) satisfies equation (1).

     An important property of equation (1) is that, for any d>0, there is at most one \lambda(d)>0 such that (1)
has a positive solution u with \lambda =\lambda(d)  and  u(0)=d.
 Let

       T={d>0 :  (1) has a positive solution with  u(0)=d },

then T is an open subset of  R⁺; \lambda(d)  is a well-defined continuous function from T  to  R⁺ .    
We call  R⁺ x R⁺ ={ (\lambda, d) |  \lambda >0, d>0 }  the  phase space,
{(\lambda (d), d): (1) has a solution with u(0)=d, \lambda=\lambda (d)} the bifurcation curve,
and the phase space with bifurcation curve the  bifurcation diagram.
 

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 Mathematical Theory from plotting

Exact Multiplicity of positive solutions of a class of semilinear problems

Exact Multiplicity of positive solutions of a class of semilinear problems: II

and the many references therein. But all these rigorous results are still very restrictive. For example, the curve with multiple turning points is verturally unsolvable for most case. So numerical calculation is a good way to get dependable information for some problems which you can never solve. Following are some suggestions for your numerical experiments:
 

Numerical Algorithm

(3)
                     u''+(n-1)/r *u'+f(u)=0,    r > 0
                     u'(0)=0, u(0)=u_0.

Notice that the new equation does not contain parameter \lambda. We use Runge-Kutta method to solve equation (3), we can get the values  of u(r), u'(r) at each point r(k)=k*h+u_0, where k is a positive integer, h is the number which you input as "Numerical mesh" and u_0 is the initial height, aslong as u(r)>0.

   The shooting method is simple: if for some k>0, u(r(k))>0 and u(r(k+1))<0, then we stop shooting, and return the value R=r(k). Then we rescale the equation (3), let \lambda(u_0)=R*R, then we get a solution of (1). The pair (\lambda(u_0), u_0) is a point which we will actrually plot on the Bifurcation Diagram.

   At this point, I have not discussed the convergence of this algorithm. It seems to be OK for most cases, even with not too small mesh size and not too many plotting points. But obviously, it fails when the value of u_0 is getting too large. I believe that there is a converge range of x_0, with given maximum error, mesh-size and initial point x_0. I hope I can theoriotically prove it later.

    If you want fast response of the applet, then you must have a trade-off between the computation-time and the accuracy of the diagram. For a better data set of (\lambda, u), we also implement the above algorithm in C++, which will result in a file containing the data set. With this program, we can take a longer time to get better result: for example, you can set the mesh size to be 0.0001 or even 0.00001, and computing with u_0 increasing each 0.01 or less. For u_0 from 0 to 20, this may take 10 minutes or so in a PC. The good part of the C++ program is

• We can know the Morse index of each solution, which is also outputed.
• We can get the data of sign-changing solutions, with the number of sign-changing inputed  by the user.
• The program is interactive, except the function, all other parameters are inputed when the program is executed.

 

 Java Applet programming

 BifurGraph.java      Equation.java       Apoint.java

and the programs provided by  Leigh Brookshaw are

parse1d.java     and the  graph package .