Samples
This software comes with the folder Samples containing numerous ready to use samples of Initial Value Problems (IVPs) in files of two kinds:
Certain scripts can be loaded directly from Demo by choosing the desired sample via a hierarchical menu – this is the recommended way for first exploration of the software.
The selection of the samples covers various branches of applied mathematics: from purely parametric curves (where the ODE section contains merely a trivial ODE t'=1) to many samples in celestial mechanics, general mechanics (single and double pendulums, rolling disk), examples of special functions such as the Bessel function (which are regular, but their ODEs are singular), and some others examples.
The full description for all of them (called Exploratorium) is in preparation.
The samples run via the main menu item Demo
Spirals ® Cornu Curls Double spiral (as an x, yextraction of the complex tangent. Each lap takes the same time to run when Played)
3 Bodies ® Lagrange case ® 1 IVP 3 IVPs
Disturbed ® 2D 3D
Periodic (closed I2A "Butterfly" chainlike orbits) VIII15B
"Spaghetti"
Periodic (open #1 free fall orbits #18 from rest points) A list of 30
Relatively periodic
Choreographies #1 (eight)
#2 . . . . . .
NonNewtonian ® Precession ® Power 1 > 1.5 (Newtonian) Power 1.7 < 1.5 (Newtonian) Choreography ® Power 1 > 1.5 (Newtonian) Power 1.7 < 1.5 (Newtonian) 4 Bodies (3D) ® Toruslike trajectories Trajectories inscribed in a cube
7 Bodies 5 Lagrange points ® m_{6} = 0.1, m_{7} = 0.9 m_{6} = 0.001, m_{7} = 0.999 Möbius outline (3D) Lorenz attractor (3D) Linked tori (3D) Rolling disk (3D) (visualized with rolling triangle, axes, orš several fixed points in it) 
These and many other samples may be opened either as ODEs or scripts from the File menu navigating to the subfolder Samples.
The folder Samples contains subfolders considered separately:
SpecPoints – containing samples of integration beginning with a singular point of ODEs yet having a regular solution, for which the Taylor expansion in at the special point was brought from other sources.
Periodic – containing samples of Choreographies and other special periodic cases of the 3 body problem.
LagrangePointsWorkshop – containing the algebraic equations whose solutions are the Lagrange points, and ODEs for the inverse functions whose integration delivers those points.
Here is the list and minimal explanations for the 72 samples in the folders. For every script file with the extension .scr there always exists the respective file .ode (not listed here if it accompanies the scr file). Go to File menu to…
1) Go to Demo/3 bodies/... and one of the
lists.
2) Enter a number of the desired sample. The
program automatically loads the ODEs for the 3 body problem with the initial
values corresponding to the selected orbit, compiles, and opens the Graph
window for visual integration. At that, the length of the period of this sample
is entered into the termination condition and the program is set for integration
until this termination point. (The period of the orbit is visible also in the
Front panel in the Constant section as a comment line for constant
a).
3) By default, in the Graph window you
will see already a result of integration of 10 steps. Typically 10 steps are
not enough for reaching the terminal point. You may need to click the button More
several times changing the number of steps to something big, say 1000. For some
samples with long periods you may need more than 1000 steps. (By default the storage
is reserved for 1000 points, therefore when the program asks you to append the
storage, do it adding say 3000 or so). You may notice that graph has not enough
room (in default setting). Any moment you can adjust the room clicking at the
button Adjust.
4) Finally when you have reached the termination
point (the period) obtaining the full graph of this orbit, you may wish to Play
it dynamically. The default time 5 sec is too short. Enter a longer interval.
Depending on the complexity of the curve, it may be something like 6080
seconds. Enjoy the show, and then repeat everything from step 1 for another
sample.
5) If you like the curves that you have obtained and played, you may wish to
save this session as the script so that next time they integrate ready for
playing automatically. In order to do this, in the label below note how
many steps n the integration took till reaching the termination point.
Enter this n under the button More, proceed to the Main Panel and
under File Save as Script into your location of choice (available only
in the licensed version). Next time you will be able to open this script and
play it.
Many of the samples below and few others are considered in a frame of teaching here
and here.
Pure parametric curves samples
CornuSpiral.scr 
Cornu spiral 
CurlySpiral.scr 
Curves looking like curls 
Mobius.scr 
Möbius outline (3D) 
MobiusLarge.scr 
Möbius outline of a larger stripe 
MobiusArray.ode 
Möbius outlineas as an array of IVPs (must be unfolded) 
Tori.scr 
Linked tori outlined with screw lines (3D) 
ToriArray.ode 
Linked tori outlined with latitudes and longitudes as an array of IVPs (must be unfolded) 
TrefoilKnot3D.scr 
Trefoil knot (3D) 
KnotChain3D.scr 
Knot chain (3D) 
CoshVsPar.scr 
The hyperbolic cosine vs. a parabola 
Mechanics
Pendulums
Pendulum2D.scr 
The mathematical pendulum 2D 
DoubePendulum.scr 
The double pendulum 2D 
PendulumApple.scr 
The mathematical pendulum 3D 
PendulumFlower.scr 
The mathematical pendulum 3D 
Rolling disk
Credit to Prof. Dmitry Garanin
As this program cannot visualize a motion of an actual disk in 3D, its motion is visualized …
Some examples: 

RollingDisk.scr 
Rolls along a straight line 
RollingDisk01.scr 
Rolls along a circular line 
RollingDisk0103.scr 
Rolls along multiple circular lines 
RollingDisk103.scr 
Rolls chaotically 
RollingDisk0103.scr 
Rolls along a semiarch 
RollingDiskTriangle.scr 
Rolling is visualized via a triangle inscribed into the disk (set Parameters/Drawing lines/TRaingle) 
RollingDiskAxes.scr 
Rolling is visualized via the 3 coordinate axes of the disk (set Parameters/Drawing lines/Axes) 
The Newtonian nbody problems
2 bodies
2BodiesProbe.scr 
the simplest 2 body problem one of which is a probe (of near zero mass) 
2BodiesProbe3D.scr 
the same as above in 3D 
2plusProbe.ode 
2 bodies and a probe (of near zero mass) in the linear formation 
2plusSmall.ode 
2 bodies and a small body in the linear formation 
3 bodies
3Bodies2D.scr 
The Lagrange case 
3Bodies2D3IVPs.scr 
The Lagrange case, 3 aggregated IVPs with different initial velocities 
3Bod9995.scr 
A slightly disturbed Lagrange case – a chaotic dance of three 
3EqBodEuler.scr 
The Euler case of 3 bodies of equal masses in linear formation 
3NonEqBodEuler.scr 
The Euler case of 3 bodies with the small central mass 
NonNewton1.7.scr NonNewton1.0.scr NonNewton0.5.scr NonNewton1.0Choreo.scr NonNewton1.7Choreo.scr 
NonNewtonian motions with the powers other than 2:  relative periodic
 Choreography 
(folder Samples/Periodic) Simo1.scr 
Scripts of 7 choreographies
from the 345 Carles Simo's list. 
(folder Samples/Periodic) I2A.scr 
Scripts of 2
periodic orbits from the list of 204 by Ana Hudomal. 
(folder Samples/Periodic) FreeFall1.scr FreeFall18.scr 
Scripts of 2
periodic free fall orbits from the list of 30 by Xiaoming Li and Shijun Liao. 
(folder Samples/Periodic) RelativePeriodic.scr 
A script of a relative periodic orbit by Ana Hudomal. 
4 bodies
4BodiesCubic.scr 
Four bodies in 3D space each having near a plane orbit inscribed in a cube (credit to Cris Moore & Michael Nauenberg) 
4BodiesTorus.scr 
Four bodies in 3D space whose orbits outline a flattened torus 
4BodiesTetra.scr 
Four bodies in 3D space placed at vertices of the regular tetrahedron. It was proved that only with radial initial velocities the formation (similar to the Lagrange case in plane) may hold (until the collision or escape) 
4BodiesTetraCollision.scr 
Four bodies in 3D space placed at vertices of the regular tetrahedron with zero velocities on collision course 
4BodiesPlane.scr 
Four bodies in plane (the Lagrange case) 
5 bodies
5Bodies3D.scr 
Five bodies in 3D in the Lagrange formation plane 
5BodiesEuler.scr 
Five bodies (including 2 probes) in 3D in the linear Euler formation in plane 
7 bodies at the 5 Lagrange points
5 probes at the 5 Lagrange points with different proportions of the masses m_{6} and m_{7}
LagrangePoints001999.scr 
m_{6} = 0.001; m_{7} =0.999 
LagrangePoints0595.scr 
m_{6} = 0.05; m_{7} =0.95 
LagrangePoints1090.scr 
m_{6} = 0.1; m_{7} =0.9 
LagrangePoints5050.scr 
m_{6} = 0.5; m_{7} =0.5 
20 bodies
20BodiesCirc.scr 
20 bodies Lagrange case circular 
20BodiesEll.scr 
five bodies Lagrange case elliptic 
NonNewtonian 3 body problems for the central force
other than the inverse square law whose exponent in the ODEs is –1.5 = 0.5 – 2
NonNewton0.5.scr 
Exponent = 0.5, precession of the orbits 
NonNewton1.0.scr 
Exponent = 1, precession of the orbits 
NonNewton1.7.scr 
Exponent = 1.7, precession of the orbits 
NonNewton1.0Choreo.scr 
Exponent = 1, orbits get closed as a choreography 
NonNewton1.7Choreo.scr 
Exponent = 1.7, orbits get closed as a choreography 
A chemical reaction
WineGlass.scr 
Oscillating chemical reactions with a graph looking like a wineglass (courtesy of the late Prof. Robert L. Borrelli, Courtney S. Coleman) 
Special functions
Bessel2.scr 
The Bessel function J_{2} from the initial value x_{0}=0.00001>0 because the Bessel ODE is singular at x=0. The initial value of the function at point x_{0} is computed (in the section Constants) employing a segment of the special Taylor expansion for J_{2 }at zero. There are more examples of the Bessel functions in the folder SpecPoints from the initial point x=0 utilizing the special expansions of J_{p} at zero. 
LorenzAttractor.scr 
Lorenz attractor (3D) 
DoubleSpiral.scr 
Double spiral is an x, y extraction of the complex tangent. Each lap takes the same time to run when Played 
Harley_t.ode HugeRad_y7x5_in_t.ode 
x^{m}/m + y^{n}/n = const for even m, n (studied by the late Prof. Harley Flanders) – "flattened circlelike" curves. They are helpful for integration with the associated ODEs in x and y. They demonstrated near infinite heuristic convergence radius. 
EqSumTo3Points_x.ode 
A curve equidistant to three points: requires switching to y in order to close the curve 
Xyyx.scr 
The commuting powers solution y(x) (satisfying x^{y}=y^{x }) vs. special hyperbola. In this example the holomorphic solution y(x) integrated from the initial point x_{0}=2 jumps over the point of singularity x=y=e of the ODE 
Holomorphic functions – solutions of singular ODEs
(folder Samples/SpecPoints)
This folder contains a selection of samples all illustrating the case when singular rational ODEs have holomorphic solutions. For some ODEs it was proven that no rational regular ODE at this special point can have those functions as solutions because the point is a special point of violation of elementariness by those functions. For some other ODEs there is no such proof, and it is not known yet whether regular rational ODEs may have these solutions at the special points – see here and here.
Bernoulli.scr 
Integration of the function t / (e^{t} 1) begins with t=0, which is possible because the special expansion at this point is provided and automatically entered (as in the samples below). 
Bessel02.scr 
In all the Bessel functions here of the index 04 integration begins with t=0. The digit "2" in the names means that the special expansion was provided for the derivative of order 2 rather than for the function proper.

Bessel12.scr 

Bessel22.scr 

Bessel32.scr 

Bessel42.scr 

Cauchyn!.ode Cauchyn!array10.scr 
This is the only example where also the solution is singular at t=0 in the Cauchy ODE t^{2}x' + (t1)x + 1 = 0 whose formal Taylor coefficient at t=0 is a_{n} = n! 
cos(sqrt(t))2.scr 
Function cos(sqrt(t)) 
InvBernoulli.scr 
Function (e^{t} 1) / t 
Ln(tplus1)divt.scr 
(Ln(t + 1)) / t 
sin(t)Divt.scr 
sin(t) / t 
x^y=y^x1.scr 
Various forms of the function of the commuting power y(x) satisfying the equation x^{y}=y^{x}

x^y=y^x2.scr 

x^y=y^xParam.scr 

x2=xDivt.scr 
Solution of the equation x'' = x/t 
z^(y^x)=(z^y)^x.scr 
Various forms of the associating powers satisfying the equation z^(y^x)=(z^y)^x 
z^(y^x)=(z^y)^x1.ode 
See more explanations here.