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