DESIR
Differential Equations Systems Integrator and Researcher
Application package DESIR (Differential Equations Systems Integrator & Research) for the investigation of differential equations systems. This package is devoted to the numerical investigation of ordinary differential equations and maps on IBM compatible PCs (MS-DOS) and is made up of a collection of executed programs and libraries for BORLAND TURBO PASCAL (versions 5.5,6.0,7.0). The programs being executed are designed to the introduction of new information to the archive or to the choice of the existing system of differential equations, to generating the program in TURBO PASCAL language for the given system of equations,to starting of the TPC translator of the version corresponding to the package libraries, and to starting of the created program of investigating the system of differential equations.
The system under study should be given in the form: dY/dt=F(t,Y,P) , where Y is the vector of N components (system's dimensionality, maximum value of which equals 8) and P is the vector of parameters. Besides the system itself, the matrix of linearized system (Jacobi matrix) should be introduced.
The main features of the program are the following:
- Construction of the trajectory of the given system starting from the initial point in different projections in the graphic windows of the screen.
- Assignment of graphic windows (quantity, demensions, coordinate axes) and of the regime of reflecting the trajectory on the screen (color, step, visibility, etc.).
- Assignment and change of the initial point and of any parameter of the system.
- Choice of the numeric method of solving the system of differential equations (possible methods: Adams, Runge-Kutta, Gear, extrapolation), of solving the eigenvalues and eigenvectors of matrix (Jacobi or HQR methods) etc.
- Search of equilibrium points (its eigenvalues, analysis of criticals cases) of the given system, of periodic solutions (cycles and its multiplicators), movement along one parameter with the search of equilibrium points and cycle bifurcation, construction of Poincare-maps, computation of the spectrum of Lyapunov and of fractal dimension,of the number of rotation of quasi-periodic regimes, construction of 3-dimensional projection, etc.
- Saving of the current state to the disk with further continuation of the research starting from the saved spot.
- Storing information about all systems studied previonsly and their states.
DESIR package can be used in research connected with ordinary differential equations (mathematical modeling in mechanics, physics, chemistry, biology, ecology) and in teaching students courses of differential equations, bifurcation theory, mathematical modeling, chaotic dynamics, etc.
Keywords | Bifurcation analysis, Dynamics (phase diagrams), Visualization |
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