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 mono-valued IVP); or they represent a multi-valued 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


Double spiral (as an x, y-extraction of the complex tangent. Each lap takes the same time to run when Played)


3 Bodies  →     Lagrange cases  →     Equal masses

                                                          Unequal masses →         Elliptic


                                                            3 IVPs

                                                            Disturbed          →        2D



                        Euler cases       →        Elliptic



Periodic (closed            I2A "Butterfly"

 chain-like orbits)       VIII15B "Spaghetti"    
                                                 . . . . .

                                     A list of 203


Periodic (open               #1

free fall orbits              #18

from rest points)           A list of 30


Relatively periodic                     


Choreographies             #1   (eight)      


                                    . . . . . .

                                    A list of 345


Non-Newtonian →       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) →         Torus-like trajectories  

                                    Trajectories inscribed in a cube


7 Bodies 5 Lagrange points →  m6 = 0.1,     m7 = 0.9

                                               m6 = 0.001, m7 = 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 60-80 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)     



Cornu spiral


Curves looking like curls


Möbius outline (3D)


Möbius outline of a larger stripe


Möbius outlineas as an array of IVPs (must be unfolded)


Möbius outline via a sine wave


Linked tori outlined with screw line (3D)



Linked tori outlined with latitudes and longitudes as an array of IVPs (must be unfolded)
Linked tori as script


  Klein bottle outlined with screw line (3D)


Klein bottle outlined with longitudes as an array of IVPs (must be unfolded)

Klein bottle as script


Pseudo Sphere (3D)


Tractricoid curve outlining pseudo sphere


Trefoil knot (3D)


Knot chain (3D)


The hyperbolic cosine vs. a parabola








The mathematical pendulum 2D


The double pendulum 2D


The mathematical pendulum 3D


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 …

  • either as 10 point outline of the circumference of a disk, 
  • or a rolling triangle plus the trajectory of the touching  point of the disk on the plane in sub-folder Samples\RigidDisk\Triangle
  • or via the three coordinate axes plus the trajectory of the touching  point of the disk on the plane in sub-folder Samples\RigidDisk\Axes,
  • or via any number of points fixed in the moving solid body plus the trajectory of the touching  point in sub-folder Samples\RigidDisk\,
  • or in the folder Samples\RigidDisk\SelectedPoints with the trajectories of the special points such as...
    • the center or the disk,
    • some point on its edge, and...
    • the trajectory of the touching  point on the plane.

A few examples:

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


The disk rolls along a fancy touch line


The disk rolls along a touch line in the straight direction


  The case of rolling called "Eurler disk"

Visualization via 12 vertices of an icosahedron in the folder Samples\RigidDisk\


   Using 12 vertices of an icosahedron for visualization

Visualization via the special points in the folder Samples\SelectedPoints


Rolls along a straight line


Rolls along a circular line


Rolls along multiple circular lines


Rolls chaotically


Rolls along a semi-arch

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\


 Dzhanibekov effect,



The Newtonian n-body problems


2 bodies



the simplest 2 body problem one of which is a probe (of near zero mass)


the same as above in 3D


2 bodies and a probe (of near zero mass) in the linear formation


2 bodies and a small body in the linear formation



3 bodies



The Lagrange case equal masses


The Lagrange case with unequal masses, elliptic, parameterized given arbitrary masses m1, m2, m3.  


The Lagrange case with unequal masses circular


The Lagrange case, 3 aggregated IVPs with different initial velocities


A slightly disturbed Lagrange case – a chaotic dance of three in 2D


A slightly disturbed Lagrange case – a chaotic dance of three in 3D


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.


The Euler case with unequal masses circular


The Euler case of 3 bodies of equal masses


The Euler case of 3 bodies with the small central mass


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






 Non-Newtonian motions with the powers other than 2:

- relative periodic


 - Choreography

(folder Samples/Periodic)


Scripts of 7 choreographies from the 345 Carles Simo's list.
All 345 of them are available as .ode files to be compiled and played as explained here.

(folder Samples/Periodic)


   Scripts of 2 periodic orbits from the list of 204 by Ana Hudomal.
  All 204 of them are available as .ode files to be compiled and played as explained here.

(folder Samples/Periodic)



   Scripts of 2 periodic free fall orbits from the list of 30 by Xiaoming Li and Shijun Liao.
  All 30 of them are available as .ode files to be compiled and played as explained here

(folder Samples/Periodic)


  A script of a relative periodic orbit by Ana Hudomal.


4 bodies



Four bodies in 3D space each having near a plane orbit inscribed in a cube (credit to Cris Moore & Michael Nauenberg)


Four bodies in 3D space whose orbits outline a flattened torus


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)


Four bodies in 3D space placed at vertices of the regular tetrahedron with zero velocities on collision course


Four bodies in plane (the Lagrange case)



5 bodies


Five bodies in 3D in the Lagrange formation plane


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 m6 and m7



m6 = 0.001; m7 =0.999


m6 = 0.05;   m7 =0.95


m6 = 0.1;     m7 =0.9


m6 = 0.5;     m7 =0.5



20 bodies



20 bodies Lagrange case circular


five bodies Lagrange case elliptic



Non-Newtonian 3 body problems for the central force

other than the inverse square law whose exponent in the ODEs is  1.5 = 0.5 – 2



Exponent = -0.5, precession of the orbits


Exponent = -1, precession of the orbits


Exponent = -1.7, precession of the orbits


Exponent = -1, orbits get closed as a choreography


Exponent = -1.7, orbits get closed as a choreography



A chemical reaction 



Oscillating chemical reactions with a graph looking like a wineglass (courtesy of the late Prof. Robert L. Borrelli, Courtney S. Coleman)



Special functions



The Bessel function J2 from the initial value x0=0.00001>0 because the Bessel ODE is singular at x=0. The initial value of the function at point x0 is computed (in the section Constants) employing a segment of the special Taylor expansion for J2 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 Jp at zero.


Lorenz attractor (3D)


Double spiral is an x, y -extraction of the complex tangent. Each lap takes the same time to run when Played



xm/m + yn/n = const  for even m, n (studied by the late Prof. Harley Flanders) – "flattened circle-like" 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.


A curve equidistant to three points: requires switching to y in order to close the curve  


The commuting powers solution y(x) (satisfying  xy=yx ) vs. special hyperbola. In this example the holomorphic solution y(x) integrated from the initial point x0=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.   



Integration of the function t / (et -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).


In all the Bessel functions here of the index 0-4 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.  








This is the only example where also the solution is singular at t=0 in the Cauchy ODE  t2x' + (t-1)x + 1 = 0 whose formal Taylor coefficient at t=0 is an = n!


 Function cos(sqrt(t))


 Function (et -1) / t


 (Ln(t + 1)) / t


 sin(t) / t


Various forms of the function of the commuting power y(x) satisfying the equation xy=yx 





 Solution of the equation  x'' = x/t


Various forms of the associating powers satisfying the equation  z^(y^x)=(z^y)^x 



 See more explanations here.