Samples
This software comes with the folder Samples containing numerous ready to use samples of Initial Value Problems (IVPs) in files of two kinds:
a) Files with particular ODEs (their extension is .ode). You should go to File/Open in order to load them. They are either ready to be compiled (representing a conventional monovalued IVP); or they represent a multivalued IVPs, which means that first you should go to Create an array of IVPs in order to unfold this shortened form into a large aggregate IVP which then may be compiled. After compilation, you will need to specify which curves to plot, and to integrate the IVP the desired number of steps.
b) Script files (their extension is .scr). You should go to File/Open script in order to load them. Script files automate the entire sequence of actions explained in (a) from compilation to plotting the solution curves (earlier integrated along a particular segment). Therefore, after completion of a script, you immediately see the plotted curves in the Graph window. Then you may wish to Play them in a real time clicking the Play button.
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 to 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 cases → Equal masses Unequal masses → Elliptic Circular 3 IVPs Disturbed → 2D 3D
Euler cases → Elliptic Circular
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) Klein bottle (3D) Rolling disk (3D) (visualized with rolling of 10 points at the edge, or triangle, or axes, or several special points fixed in it) Dzhanibekov effect (3D) 
These and many other samples may be opened either as the ode or script files from the File menu navigating to the subfolder Samples.
The folder Samples contains numerous samples of the ode and scr files, and 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 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.
RigidDisk – containing scripts visualizing rolling of rigid disk under different conditions and different methods such as 10 points at the edge, or a triangle, or the axes, or several special points fixed in it.
Here is the list and minimal explanations for over 70 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…
Actions needed for integrating the samples from the special Lists in the three body problems
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 (blindly)
integrates the problem until reaching the termination point – the period of this simulation entered
from its file. (The period of
the orbit is visible also in the
Front panel in the Constant section as a comment line for
constant
a).
3) When the integration reaches the
termination
point (the period), the program displays the message. As you click OK,
the program opens the Graph window displaying the entire trajectory.
You may wish to Play
it dynamically. 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.
4) 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 the File menu, Save Script into a location of your
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
(as exercises in analytic geometry)
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) 
MobiusSineOutline.scr 
Möbius outline via a sine wave 
Tori.scr 
Linked tori outlined with screw line (3D) 
ToriArray.ode

Linked tori
outlined with latitudes and longitudes as an array of IVPs (must be
unfolded) 
ScrewLineKleinBottle.scr 
Klein bottle outlined with screw line (3D) 
KleinBottleForArray.ode 
Klein bottle outlined with longitudes as an array of IVPs (must be unfolded) Klein bottle as script 
PseudoSphere.scr 
Pseudo Sphere (3D) 
Tractricoid.scr 
Tractricoid curve outlining pseudo sphere 
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 
Rigid Body
See an extensive Exploratorium for the rigid body
Credit to Prof. Dmitry Garanin
As this program cannot visualize a motion of an actual rigid body in 3D, its motion is visualized …
A few examples: 

Visualization via 10 points at the edge of the Rolling disk in the folder Samples\RigidDisk\ 

RollingDisk10Points.scr 
The disk rolls along a fancy touch line 
RollingDiskStraight10Points.scr 
The disk rolls along a touch line in the straight direction 
EulerDisk10Points08.scr 
The case of
rolling called "Eurler disk" 
Visualization via 12 vertices of an icosahedron in the folder Samples\RigidDisk\ 

RollingSphereIcosahedron.ode 
Using 12 vertices of an icosahedron for visualization 
Visualization via the special points in the folder Samples\SelectedPoints 

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 
Visualization via a triangle of the above samples in the folder Samples\Triangle 

Visualization via the axes of the above samples in the folder Samples\Axes 

Free rigid body, folder Samples\FreeTop\ 

L2onlyAsTriang.scr 
Dzhanibekov
effect, 
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 equal masses 
Lagrange.scr 
The Lagrange case with unequal masses, elliptic, parameterized given arbitrary masses m1, m2, m3. 
LagrangeCircular.scr 
The Lagrange case with unequal masses circular 
3Bodies2D3IVPs.scr 
The Lagrange case, 3 aggregated IVPs with different initial velocities 
3Bod9995.scr 
A slightly disturbed Lagrange case – a chaotic dance of three in 2D 
3Bodies3D.scr 
A slightly disturbed Lagrange case – a chaotic dance of three in 3D 
Euler.scr 
The Euler case with unequal masses elliptic, parameterized. Given the positions 0, 1, 1+r and arbitrary masses m1, m2, the mass m3 is determined. 
EulerCircular.scr 
The Euler case with unequal masses circular 
3EqBodEuler.scr 
The Euler case of 3 bodies of equal masses 
3NonEqBodEuler.scr 
The Euler case of 3 bodies with the small central mass 
Slingshot.scr 
The K. Sitnikov case. The three body case when a couple of them is engaged into elliptic Kepler motion in a plane, while the 3rd one runs along a perpendicular through the center of masses in a manner of a slingshot 
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 (Harley_x.ode, Harley_y.ode). 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 
xyyxJumpsOverSing.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.