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 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
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Spirals → Cornu Curls 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 Circular 3 IVPs Disturbed → 2D 3D
Euler cases → Elliptic Circular
Periodic (closed I2A "Butterfly" chain-like orbits)
VIII15B "Spaghetti"
Periodic (open #1 free fall orbits #18 from rest points) A list of 30
Relatively periodic
Choreographies #1 (eight)
#2 . . . . . .
Non-Newtonian central force rn, i15 = (n - 1)/2, Newtonian n = -2, i15 = -1.5 Precession → i15 = -1 > -1.5, n = -1 i15 = -1.7 < -1.5, n = -2.4 Choreography → i15 = -1 > -1.5, n = -1 i15 = -1.7 < -1.5, n = -2.4 Linear spring → i15 = 0, n = 1
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 into a
small window (top left). 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 2 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)
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CornuSpiral.scr |
Cornu spiral |
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CurlySpiral.scr |
Curves looking like curls |
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Mobius.scr |
Möbius outline (3D) |
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MobiusLarge.scr |
Möbius outline of a larger stripe |
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MobiusArray.ode |
Möbius outlineas as an array of IVPs (must be unfolded) |
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MobiusSineOutline.scr |
Möbius outline via a sine wave |
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Tori.scr |
Linked tori outlined with screw line (3D) |
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ToriArray.ode
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Linked tori outlined with
latitudes and longitudes as an array of IVPs (must be unfolded) |
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ScrewLineKleinBottle.scr |
Klein bottle outlined with screw line (3D) |
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KleinBottleForArray.ode |
Klein bottle outlined with longitudes as an array of IVPs (must be unfolded) Klein bottle as script |
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PseudoSphere.scr |
Pseudo Sphere (3D) |
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Tractricoid.scr |
Tractricoid curve outlining pseudo sphere |
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TrefoilKnot3D.scr |
Trefoil knot (3D) |
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KnotChain3D.scr |
Knot chain (3D) |
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CoshVsPar.scr |
The hyperbolic cosine vs. a parabola |
Pendulums
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Pendulum2D.scr |
The mathematical pendulum 2D |
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DoubePendulum.scr |
The double pendulum 2D |
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PendulumApple.scr |
The mathematical pendulum 3D |
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PendulumFlower.scr |
The mathematical pendulum 3D |
Rigid Body
See an extensive Exploratorium for the rigid body
Credit to Prof. Dmitry Garanin
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As this program cannot visualize a motion of an actual rigid body in 3D, its motion is visualized
A few examples: |
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Visualization via 10 points at the edge of the Rolling disk in the folder Samples\RigidDisk\ |
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RollingDisk10Points.scr |
The disk rolls along a fancy touch line |
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RollingDiskStraight10Points.scr |
The disk rolls along a touch line in the straight direction |
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EulerDisk10Points08.scr |
The case of rolling called
"Eurler disk" |
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Visualization via 12 vertices of an icosahedron in the folder Samples\RigidDisk\ |
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RollingSphereIcosahedron.ode |
Using 12 vertices of an icosahedron for visualization |
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Visualization via the special points in the folder Samples\SelectedPoints |
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RollingDisk.scr |
Rolls along a straight line |
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RollingDisk01.scr |
Rolls along a circular line |
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RollingDisk01-03.scr |
Rolls along multiple circular lines |
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RollingDisk1-03.scr |
Rolls chaotically |
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RollingDisk0103.scr |
Rolls along a semi-arch |
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Visualization via a triangle of the above samples in the folder Samples\Triangle |
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Visualization via the axes of the above samples in the folder Samples\Axes |
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Free rigid body, folder Samples\FreeTop\ |
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L2onlyAsTriang.scr |
Dzhanibekov effect, |
The Newtonian n-body problems
In all problems for Newtonian motion, there is a special constant i15 used in the equations
(this constant is neg1p5 in auto-generated n-body problems).
Presuming that the central force is proportional to rk, i15 = (k 1)/2.
For the Newtonian central force, k = 2 so that i15 = 1.5.
For non-Newtonian central force i15 differs from 1.5.
In particular, for a linear spring k = 1 so that i15 = 0.
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2 bodies
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2BodiesProbe.scr |
the simplest 2 body problem one of which is a probe (of near zero mass) |
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2BodiesProbe3D.scr |
the same as above in 3D |
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2plusProbe.ode |
2 bodies and a probe (of near zero mass) in the linear formation |
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2plusSmall.ode |
2 bodies and a small body in the linear formation |
3 bodies
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3Bodies2D.scr |
The Lagrange case equal masses |
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Lagrange.scr |
The Lagrange case with unequal masses, elliptic, parameterized given arbitrary masses m1, m2, m3. |
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LagrangeCircular.scr |
The Lagrange case with unequal masses circular |
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3Bodies2D3IVPs.scr |
The Lagrange case, 3 aggregated IVPs with different initial velocities |
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3Bod9995.scr |
A slightly disturbed Lagrange case a chaotic dance of three in 2D |
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3Bodies3D.scr |
A slightly disturbed Lagrange case a chaotic dance of three in 3D |
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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. |
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EulerCircular.scr |
The Euler case with unequal masses circular |
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3EqBodEuler.scr |
The Euler case of 3 bodies of equal masses |
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3NonEqBodEuler.scr |
The Euler case of 3 bodies with the small central mass |
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Slingshot.scr SlingshotZeroMass.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 |
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NonNewton-1.7.scr NonNewton-1.0.scr NonNewton-0.5.scr NonNewton-1.0Choreo.scr NonNewton-1.7Choreo.scr NonNewton0.scr |
Non-Newtonian motions with the powers other than 2: - relative periodic
- Choreography
Linear spring |
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(folder Samples/Periodic) Simo1.scr Simo1w1w2w3.scr |
Scripts of 7 choreographies
from the 345 Carles Simo's list.
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(folder Samples/Periodic) I2A.scr |
Scripts of 2
periodic orbits from the list of 204 by Ana Hudomal. |
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(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. |
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(folder Samples/Periodic) RelativePeriodic.scr |
A script of a relative periodic orbit by Ana Hudomal. |
4 bodies
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4BodiesCubic.scr |
Four bodies in 3D space each having near a plane orbit inscribed in a cube (credit to Cris Moore & Michael Nauenberg) |
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4BodiesTorus.scr |
Four bodies in 3D space whose orbits outline a flattened torus |
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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) |
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4BodiesTetraCollision.scr |
Four bodies in 3D space placed at vertices of the regular tetrahedron with zero velocities on collision course |
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4BodiesPlane.scr |
Four bodies in plane (the Lagrange case) |
5 bodies
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5Bodies3D.scr |
Five bodies in 3D in the Lagrange formation plane |
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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 m6 and m7
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LagrangePoints001-999.scr |
m6 = 0.001; m7 =0.999 |
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LagrangePoints05-95.scr |
m6 = 0.05; m7 =0.95 |
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LagrangePoints10-90.scr |
m6 = 0.1; m7 =0.9 |
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LagrangePoints50-50.scr |
m6 = 0.5; m7 =0.5 |
20 bodies
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20BodiesCirc.scr |
20 bodies Lagrange case circular |
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20BodiesEll.scr |
five bodies Lagrange case elliptic |
Non-Newtonian 3 body problems for the central force
(i15 ≠ -1.5)
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NonNewton-0.5.scr |
i15 = -0.5, precession of the orbits |
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NonNewton-1.0.scr |
i15 = -1, precession of the orbits |
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NonNewton-1.7.scr |
i15 = -1.7, precession of the orbits |
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NonNewton-1.0Choreo.scr |
i15 = -1, orbits get closed as a choreography |
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NonNewton-1.7Choreo.scr |
i15 = -1.7, orbits get closed as a choreography |
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NonNewton0.scr |
i15 = 0, linear spring |
A chemical reaction
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WineGlass.scr WineGlassx(t)y(t).scr |
Autocatalators chemical reactions with a graph looking like a wineglass (courtesy of the late Prof. Robert L. Borrelli, Courtney S. Coleman) |
Special functions
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Bessel2.scr |
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. |
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LorenzAttractor.scr |
Lorenz attractor (3D) |
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DoubleSpiral.scr |
Double spiral is an x, y -extraction of the complex tangent. Each lap takes the same time to run when Played |
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Harley_t.ode HugeRad_y7x5_in_t.ode |
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. |
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EqSumTo3Points_x.ode |
A curve equidistant to three points: requires switching to y in order to close the curve |
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xyyxJumpsOverSing.scr |
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 |
Pure Mathematics
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, here.
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Bernoulli.scr |
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). |
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Bessel-0-2.scr |
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.
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Bessel-1-2.scr |
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Bessel-2-2.scr |
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Bessel-3-2.scr |
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Bessel-4-2.scr |
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Cauchyn!.ode Cauchyn!array10.scr |
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! |
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cos(sqrt(t))-2.scr |
Function cos(sqrt(t)) |
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InvBernoulli.scr |
Function (et -1) / t |
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Ln(tplus1)divt.scr |
(Ln(t + 1)) / t |
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sin(t)Divt.scr |
sin(t) / t |
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Commuting and Associating Powers
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x^y=y^x1.scr |
Various forms of the function of the commuting power y(x) satisfying the equation xy=yx
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x^y=y^x2.scr |
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x^y=y^xParam.scr |
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x2=xDivt.scr |
Solution of the equation x'' = x/t |
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z^(y^x)=(z^y)^x.scr |
Various forms of the associating powers satisfying the equation z^(y^x)=(z^y)^x |
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z^(y^x)=(z^y)^x1.ode |
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A remarkable periodicity in a real-valued extraction of a well-known complex function
(folder Samples\DoubleSpiralStudy)
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CircSegment.scr DoubleSpiral.scr SteepSpiral.scr Verification.scr VerificationAsympt.scr VerificationAsympt1.scr 4DoubleSpirals.scr 4DoubleSpiralsBig.scr 5DoubleSpirals.scr CircleODEs.scr |
The simulations for the article (listed in the Appendix 2) |
See more explanations here.