Dynamics Solver Tutorial

This page shows briefly what Dynamics Solver can do. Only some of
the most basic possibilities of the program will be discussed here. In the
manual and the help file, the user can find both a complete description of each
feature and a systematic discussion of the way in which a problem can be set up
and input into Dynamics Solver.

Chaotic
Systems

Using
Projections and Poincaré Sections

Using
Constrained Initial Conditions and Boundary Conditions

Numerical
Simulation

Discrete
Dynamical Systems

Function Graphs
and Parametric Curves

Drawing with
Dynamics Solver

Decorative
Examples

Chaotic systems

Dynamics Solver can easily solve the famous Lorenz equations. You will see
different projections of a beautiful orbit that stays forever in a well-defined
region, but never repeats itself.

The orbit wanders around a complicated object (a so-called strange attractor)
in the three-dimensional phase space, but you only see its projections on the
different coordinate planes and one of the windows shows a projection along a
view line.

Using Projections and Poincaré sections

To get a better idea of the shape of a three-dimensional orbit, you can use
the projections on the three coordinate planes, (x,y),
(y,z) and (x,z), as shown in three of the windows in

In Dynamics Solver you can also project the solution (in perspective or not)
along an arbitrary direction, as shown in another window. You can also see in
the figure that the section of the torus is not circular. Moreover, you can
directly see this section in the window entitled Poincaré section where
the (x,z) coordinates of the points of intersection of the orbit
with the y = 0 plane are displayed.

This intersection of the orbit with a plane is a very specific Poincaré
section, but Dynamics Solver is able to compute rather general Poincaré
sections. More interesting examples are computed in other problem files. For
instance, one can see how works the "stretch-and-fold" mechanism in the strange
attractor of Duffing equation

Furthermore, the same basic mechanism used by Dynamics Solver to draw
Poincaré sections can be used to compute the periods of closed orbits and other
quantities.

Using Constrained Initial Conditions and Boundary
Conditions

Projections and Poincaré sections are useful to obtain two-dimensional
information from higher-dimensional systems. There is still another way to lower
the dimension. In a Poincaré section one chooses from all the solutions only
those points that satisfy a given condition. It is also possible to choose all
the points, but only from those solutions that satisfy a certain condition. In
Dynamics Solver, this can be done by imposing a set of conditions on the initial
values of the problem. This allows analyzing subspaces of the full phase space
of the problem.

For example, consider the Hénon-Heiles system. The energy is a conserved
quantity. If you are interested only in those solutions corresponding to a given
value of the energy, you can choose at will the initial values for x,
y and dy/dt, for instance, and then use the energy value
and its definition to select dx/dt. You can easily instruct
Dynamics Solver to do that automatically. The solutions corresponding to a
certain energy value span the so-called "energy surface," which has three
dimensions. One can then take a Poincaré section of this energy surface:

Constrained initial conditions are useful not only to analyze energy surfaces
and more general subspaces of the phase space of a problem, but also to solve in
Dynamics Solver complex problems that do not appear in one of the simple forms
discussed so far. Note also that, though only a single Poincaré condition can be
imposed in Dynamics Solver, you can impose one additional condition for each
initial value. These conditions need not be constants of motion of the problem.
It is also possible to use constrained conditions which are not solved for the
initial values.

Finally, a very general class of boundary conditions may be handled by
Dynamics Solver. Run the example in Examples\Quantum\spectrum1.ds to
obtain numerically the well-known energy spectrum and wave functions of the
quantum harmonic oscillator:

Numerical simulation

Dynamics Solver is a good tool for numerical simulation of complex systems.
For instance, by sending multiple output to a single window you may see how
evolve three stars under their mutual gravitational attraction. A classical
example is solved in Examples\Mechanics\Burrau.ds:

You may also see in real time how chaotic scattering happens in
Examples\Chaos\disks2.ds:

Discrete Dynamical Systems

Dynamics Solver can also solve many functional differential equations. and
many discrete dynamical systems, which can be obtained by iterating maps in
arbitrary dimensions or by using more general recurrence relations. Among other
possibilities, one can compute and display their evolution, bifurcation
diagrams, histograms, cobweb diagrams and Liapunov coefficients.

Many examples are include in the Examples\Art and
Examples\Chaos directories. Let us only mention here that one can
display, among other things:

1. Different magnifications of strange attractors:

2. Bifurcation diagrams, including Liapunov exponents:

3. Single and double cobweb diagrams:

4. More sophisticated graphics, as the "devil's staircase" displayed below:

Function Graphs and Parametric Curves

You can also use Dynamics Solver to plot the graph of a function (Euler
Gamma function in the graph below) by solving a trivial differential
equation:

Parametric curves in the plane are also readily obtained in a similar way.
These are Lissajous curves:

It is also possible to draw projections of three-dimensional curves.
Lissajous figures in three dimensions are displayed below:

Drawing with Dynamics Solver

Dynamics Solver can be used to draw rather complex figures including
segments, circles, ellipses, parametric curves in two and three dimensions, text
strings (with Latin, Greek and Cyrillic letters), arrowheads, points and lines
from external data files and a large class of fractal curves. This ability is
very useful to add lettering and other informative elements to a problem
solution, as illustrated in different example problem files, such as
Examples\ODEs\autocatalator.ds.

or when preparing drawings for courseware material and research papers:

Several interesting examples are included in the Examples\Drawing
and Examples\Fractal directories. Let us mention here another (better)
way to draw Lissajous figures (Examples\Drawings\Lissajous1.ds):

and a typical fractal curve (Examples\Fractal\snowflake.ds):

Decorative Examples

Dynamics Solver can also be used to get some artistic (!?) drawings as in
many examples included in the Examples\Art and Examples\Chaos
directories:

Animations

Dynamics Solver may also be used to produce simple animations of
physical or mathematical systems. The frames may be captured and then an
external program may use them to create a video file or an animated GIF picture,
such as the examples of the evolution of a double pendulum in
Examples\Mechanics\double1.ds

double.avi

and a combination of the harmonic spring and the pendulum in
Examples\Mechanics\spring-pendulum1.ds

Azken aldaketa: 2006-09-01

Copyright © 1992-2006, Juan M.
Aguirregabiria
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