Applications of the Taylor Center software
for teaching and research

This page reviews a few articles which demonstrate how this software may be used for teaching and research purposes due to its particular features such as...

- Availability of Taylor expansions and n-order derivatives of the solution;
- Real time animation of the motion along the trajectories in 2D and 3D;
- Plotting phase portraits and families of
Initial Value Problems,
and others.

Articles on application
of the Taylor Center software for teaching

The three articles below exemplify various teaching ideas and topics for which the Taylor Center software happened to be beneficial or indispensable.

1. How the Taylor Center may assist in
teaching mathematics (by A. Gofen, 2011)

This article addressed a variety of topics beginning with the most straightforward feature...

* Visualization of dynamics via real time drawing. A selection of the analyzed examples are...

- A chemical reaction (a "vine glass" - credit to the late Prof. Borrelli);
- Double pendulum with its chaotic movement;

- Other illustrative examples offered in the Demo menu and Script files, including...

- The Lagrange case of the Three-body problem, used as a
platform for and demonstration for a few other related topics such
as...

- Instability of the Lagrange case (demonstration of a
beautiful disturbed case) in 2D and 3D;

- Choreography of n-body problem;

- Posing a question whether an analogue of the Lagrange case is possible in 3 dimensions on the vertices of the classical 5 regular polyhedra rather than on the regular polygons in 2 dimensions. The answer is No, and the proof is provided in the Appendix

- Behavior of the Taylor terms as the bell shape, visualized in the Dynamic Profile Diagrams demonstrating how the step affects the bulge;
- Heuristic and true convergence radii. How the program compute them and behaves for in case of a finite and infinite convergence radius;

* Regular solutions of singular ODEs. This chapter brings and explains the idea that sometimes a regular holomorphic solution may satisfy ODEs having singularity at the point where the solution is regular. Moreover, in some functions there are regular points at which the function may satisfy only singular ODEs (in the class of rational or elementary ODEs). The demo role of the program is based on its feature to cope with ODEs having an isolated point of singularity which however is a regular point for the solution.

* Weird examples of real valued solutions vs. their complex properties. Here (among others) is analyzed the function cos(sqrt(t)) which is regular at t=0 (when properly defined). It is explained how to integrate and plot this function for t<0 despite that it involves complex values.

Gofen, Alexander (2012) "Using the Taylor Center to Teach ODEs," CODEE Journal: Vol. 9, Article 6.

Available at: http://scholarship.claremont.edu/codee/vol9/iss1/6

2. A remarkable periodicity in a real
valued extraction of a well known complex function (by A.
Gofen, S. Lucas, J. Sochacki, 2014).

This articled was prompted by a real life situation, when the authors used the Taylor Center for plotting the solution of a suggested by somebody system of ODEs (not yet knowing the origin of these equations):

*x' = x*²* *–* y*²*+
2xy *–* x *– *3.5y +
1, x(0)=0.1
y' = -x*²

however what attracted attention of the authors was a remarkable dynamic of evolution of this double spiral, animated by the program in real time. It seemed that each loop of the double spiral (no matter big or small) took the same time to run! (See this example under Demo/Spirals/Double spiral). The program allowed to accurately compute the time spans taken by the loops - and they happened to be equal (up to the rounding error), confirming that the initial impression was based on the reality.

This remarkable property (figured out by mere observation of the dynamics of drawing) doubled up my interest for examining the origin and properties of this system, which happened to be a real valued form of a complex ODE

Therefore this research exemplifies a mathematical study triggered by observation of numerical experiments at a computer, followed by the conventional mathematical analysis of the observed phenomenon. Not only did this mathematical analysis succeed in establishing the proof and explanation for what was observed, but it also revealed existence of the critical asymptotic curve. This curve would have been impossible to find by numerical experiment only. The numerical experimenting and the analytic approaches complemented each other beautifully in discovering a remarkable property of a particular planar polynomial system of ODEs - presenting a powerful teaching moment.

3. Powers which commute or
associate as solutions of ODEs (by Gofen, 2013)

This paper is dedicated to the two classical transcendental functions: The locus of points for which powers commute, and the locus of points for which powers associate. These classical functions however are considered in a new perspective: as holomorphic solutions of ODEs yet passing over the points of singularity of these ODEs.

For example, for the function y(x) of commuting powers defined by the equation x

y/x – ln y

y' = ––––––––

x/y – ln x

y' = ––––––––

x/y – ln x

or

y"x^{2}y^{2}(y – x) – (y')^{3}x^{4} + (y')^{2}yx^{2}(3x – 2y) + y'y^{2}x(3y – 2x) – y^{4} = 0 .

Both ODEs have a singular point at (e,e) though the solution y(x) is holomorphic at this point. In order to obtain the Taylor expansion for y(x) at (e,e) the standard AD formulas are not applicable because of the singularity of these ODEs, so that the Taylor coefficients at this point must be obtained via special recursive formulas. As soon as these coefficients are available, the Taylor Center becomes helpful due to its capability to start integration even at a singular point of the ODEs if the solution is regular at this point and the special Taylor expansion of the solution is provided (from other sources). Thanks to this feature, the software can deal with the special ODEs studied here and pre-loaded into the distribution package so that the users can reproduce all the graphs referred in this article.

Published in: Teaching Mathematics and Computer Science, 11/2, 2013, p. 241-254.

Exploratorium
(didactical
comments for the preloaded problems)

The software comes with the
folder Samples containing
numerous ready to use samples of Initial Value
Problems (IVPs) helpful for teaching.

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. Here is the list
and minimal explanations for several hundreds samples pre-loaded with
the program including ...

- Exploratorium
of Choreograhies for the planar Newtonian 3-body problem with equal
masses, where you will find the explanation of the concept, the
resources, and
how to run 345 samples of Choreographies kindly submitted by Prof.
Carles Simò.

- Exploratorium of periodic
solutions recently discovered for the planar Newtonian 3-body
problem with equal masses. The first colletion of 203
represents closed curves
whose initial vectors were kindly provided thanks to Ana Hudomal. The second collection
of 30 samples are the cases where the periods are represented with finite curved segments at whose
extremes the bodies have zero velocities (i.e. they are in a free fall)
- thanks to the data by Xiaoming
LI and Shijun LIAO.

The full description for all them (called Exploratorium) is in preparation.

Examples
of applications and
articles in scientific Delphi with a teaching potential

1. From Pascal to Delphi to Object Pascal-2000. ACM SIGPLAN Notices, Vol. 36, No. 6, pp. 38-49 (2001).

2. Object vs. Class: Fewer Pointers, Less Double Thinking. Delphi Informant Magazine, Vol. 5, No. 7, pp. 47-52 (1999).

An in-depth discussion about the direct (one-to-one mapped) variables vs. indirect and separated reference to variables in programming languages.

3. Dynamic Arrays. Delphi Informant Magazine, Vol. 6, No. 2, (2000).

An evolution of the concept "dynamic array" from ALGOL-60 to Delphi

4. Recursion Excursion. Delphi Informant Magazine, Vol. 6, No. 8, pp. 30-38 (2000).

Theory and examples of code on:

- parsing of arithmetic expressions into a Reverse Polish notation (RPN);
- recursive evalutation of the PRN;
- a code of an advanced calculator of expressions.

Theory and examples of code on:

- parsing of arithmetic expressions into a Reverse Polish notation (RPN);
- compiling RPN into a list of Automatic Differentiation (AD) instructions;
- a software
emulator of the AD processor processing the list of AD instructions.

7. Do-It-Yourself 3D. Delphi Informant Magazine. Vol. 10, No. 8, pp. 17-22, (2004)

Theory and examples of code on:

- geometry of stereo pair;
- anagliph (red/blue) technique of viewing stereo pair;
- implementation of the red/blue technique of viewing stereo pair using the graphic power of Delphi.

More
topics with teaching potential contained in the Taylor Center software

- The algorithm of plotting parametric 2D and 3D curves point by point based on their Taylor expansions guaranteeing their continuity.
- The algorithm of tube-like plotting 3D curves with resolution of skewing controversies in 3D stereo and in isometric plotting.
- The algorithm of real time playing trajectories according to their parametric representation as functions of time.
- The algorithm for obtaining ODEs in another state (in another independent variable) and switching the integration from one independent variable to another.