Starting diary file "diary-2016-02-24.txt" on Wed Feb 24 2016 12:29 PM.

edit ode23tx
help meshgrid
 MESHGRID   X and Y arrays for 3-D plots.
    [X,Y] = MESHGRID(x,y) transforms the domain specified by vectors
    x and y into arrays X and Y that can be used for the evaluation
    of functions of two variables and 3-D surface plots.
    ...

[y1,y2] = meshgrid(1:4)
y1 =
     1     2     3     4
     1     2     3     4
     1     2     3     4
     1     2     3     4
y2 =
     1     1     1     1
     2     2     2     2
     3     3     3     3
     4     4     4     4

[y1,y2] = meshgrid(-15 : .5 : 15);
whos
  Name       Size            Bytes  Class     Attributes

  y1        61x61            29768  double              
  y2        61x61            29768  double              

altitude = (y1/1000) .^ 2 + y2 .^ 2;

max(max(altitude))
ans =
  225.0002

min(min(altitude))
ans =
     0

surf(y1, y2, altitude)

edit valleyf

ode23(@valleyf, [0 10], [10 ; 0])
[T, Y] = ode23(@valleyf, [0 10], [10 ; 0]);
T
T =
         0
    0.8000
    1.8000
    2.8000
    3.8000
    4.8000
    5.8000
    6.8000
    7.8000
    8.8000
    9.8000
   10.0000
Y
Y =
   10.0000         0
    9.2000         0
    8.2000         0
    7.2000         0
    6.2000         0
    5.2000         0
    4.2000         0
    3.2000         0
    2.2000         0
    1.2000         0
    0.2000         0
         0         0

nsteps = length(T)-1
nsteps =
    11

close all
surf(y1, y2, altitude); hold on; plot(Y(:,1),Y(:,2),'r.'); shg

[T, Y] = ode23(@valleyf, [0 10], [10 ; 0.1]);
nsteps = length(T)-1
nsteps =
        1970

close all
surf(y1, y2, altitude); hold on; plot(Y(:,1),Y(:,2),'r.'); shg

T
T =
         0
    0.0080
    0.0482
    0.0622
    ...
   10.0000

Y
Y =
   10.0000    0.1000
    9.9992    0.0920
    9.9935    0.0522
    ...
    0.0741    0.0000
    0.0739   -0.0000
    0.0738    0.0000
    0.0736   -0.0000
    0.0735   -0.0000

Y(1:10, :)
ans =
   10.0000    0.1000
    9.9992    0.0920
    9.9935    0.0522
    9.9905    0.0385
    9.9863    0.0252
    9.9828    0.0179
    9.9782    0.0112
    9.9733    0.0069
    9.9686    0.0043
    9.9610    0.0019

Y(1:50, :)
ans =
   10.0000    0.1000
    9.9992    0.0920
    9.9935    0.0522
    ...
    9.2264   -0.0000
    9.2013    0.0000
    9.1778   -0.0000

format long
Y
Y =
  10.000000000000000   0.100000000000000
   9.999166601563283   0.092003434105903
   9.993521799027876   0.052210206767652
   9.990492313954952   0.038549256053989
   9.986258441366719   0.025217691569863
   9.982826392168530   0.017878919803373
   9.978198647088441   0.011239256616668
   9.973323144384104   0.006887738540956
   9.968640747879142   0.004300151976987
   9.960951766113295   0.001945609671813
   9.956376933928373   0.001225902956879
   ...
   0.109167436300915   0.000001030123314
   0.108893358280143  -0.000001030123310
   0.108619968353986   0.000001030123306
   0.108347264794898  -0.000001030123302
   0.108075245879672   0.000001030123298
   0.107803909889426  -0.000001030123294
   0.107533255109596   0.000001030123290
   0.107263279829918  -0.000001030123286
   ...
   0.074874596397239   0.000001030122408
   0.074686612949174  -0.000001030122399
   0.074499101444091   0.000001030122391
   0.074312060697107  -0.000001030122382
   0.074125489526318   0.000001030122373
   ...

[T, Y] = ode23s(@valleyf, [0 10], [10 ; 0.1]);
nsteps = length(T)-1
nsteps =
    43

close all
surf(y1, y2, altitude); hold on; plot(Y(:,1),Y(:,2),'r.'); shg

edit ode23s

clear

f = @(t,y) (y^2 - y^3);

delta = .01;
ode23(f,[0 2/delta], delta)

delta = .001;

ode23(f,[0 2/delta], delta)

delta = .00001;
ode23(f,[0 2/delta], delta)
ode23s(f,[0 2/delta], delta)

diary off