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Dynamical Software I nad II
SOFTWARE
REVIEW
Dynamical Software I and II, for IBM PC or compatible with graphics and Microsoft or Ryan-McFarland FORTRAN compiler. Dynamical Systems, Post Office Box 35241, Tucson, Arizona 85740; (601) 825-1331. I, $250; II, $350; both $550.
Since 1963, when E. Lorenz described chaotic oscillations in a simple threedimensional model of thermoconvection, dynamical chaos and strange attractors have become extremely popular objects in mathematics and applied sciences. Many mathematicians, physicists, and researchers from other fields united their efforts in developing a theory of strange attractors. This theory produced an explosion of interest in computer simulation and forced many “pure” mathematicians to treat computers as a necessary and natural research tool. Numerical programs for nonlinear dynamics appearing in the last decade have, in turn, made these beautiful and highly developed theories accessible to many applied researchers. In the early 198Os, F. Takens and coworkers proposed a simple procedure to study dynamical chaos through experimental time series that does not assume the existence of a mathematical model of the considered process. The main idea is to introduce an appropriate time lag and to embed the original univariate time series into a two-, three-, or higher-dimensional space. The result is a trajectory in an m-dimensional space similar to a phase trajectory of a dynamical system, which then gives a graphical image of experimental chaotic oscillations in the form of an attractor comparable to known attractors or to the attractor in a relevant model. If some essential properties of the attractor (e.g., its fractal dimension) stabilize while we increase the embedding dimension, a finite-dimensional model of the process becomes possible. The package Dynamical Software (DS) is a toolkit for studying nonlinear dynamics and chaotic oscillations on an IBM PC or PC-compatible computer. DS is equally useful with a mathematical model or only experimental data. A model may be in the form of ordinary or delay differential equations or discrete maps. Using DS you can compute and draw the trajectories for any model and calculate Lyapunov exponents as well as smoothed power spectra for ODEs. DS has two procedures for bifurcation analysis of discrete maps. It can compute and plot a bifurcation diagram with which you can keep track of the evolution of the asymptotic state when a control parameter changes with a given step. One can also plot the rotation numbers against the control parameter. These procedures are the only methods of bifurcation analysis that DS supports. The bulk of the DS package is made up of programs for studying a computed trajectory or a trajectory produced from a univariate time series by means of the
MATHEMATICAL
BIOSCIENCES
98:145-147
OElsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010
(1990)
145 0025-5564/90/$03.50
146
SOFTWARE
REVIEW
embedding procedure. This part includes common numerical tools such as smoothing and interpolation, approximation of invariant measure by a histogram, statistics, and plotting a trajectory in different forms (two-dimensional and three-dimensional projections, color-valued velocity or an additional variable, and so on). The other procedures are more complex and time-consuming algorithms such as calculation of power spectra (periodogram, autocovariance function, and smoothed spectra), estimation of fractal dimension by correlation dimension, and computation of the maximal Lyapunov value from a time series. DS is distributed in two self-contained versions. As the authors write, DSI includes basic tools while DS II includes more advanced algorithms. The main features are simulation of mathematical models (I and II), phase portraits (I), discrete maps from time series (I), smoothing and interpolation (I), spectral analysis (II), fractal dimension (II), eigenvalues and Lyapunov exponents (II), transformations and statistics (II), and graphic utilities (I and II). The simple organization of DS enables the adding of new algorithms in the future. The DS manual is very detailed and contains a theoretical introduction to dynamical chaos that is illustrated by figures produced by DS, a description of the package, and many interesting examples, including the original experimental data from biology and medicine. Some of these examples are provided on the distribution disks on the form of tests and samples. Of the 20 distribution diskettes, probably half contain examples, graphics, or data sets. Unfortunately, many important details about the numerical algorithms used have been left out of the framework of the manual. Some interactive features and special graphic facilities are necessary because numerous simulations can be needed for a proper choice of parameters. DS supports many graphics adapters including CGA, EGA, and Hercules (but not VGA) and provides a convenient means for two- and three-dimensional plots. The weakest feature of the DS interface is the question-and-answer style it uses. You have to react to many prompts by typing various numbers that must be kept in mind. It is interesting that the same numerical codes result in different actions in different DS programs. Computation and plotting of results are usually done separately. Dynamical Software is only a software package, not an integrated environment, which is now becoming standard in the personal computer world. The DS user deals with the DOS operating system, files, a FORTRAN compiler, and linker. To make using DS more pleasant, the authors have provided some batch files and an on-line manual. An important useful feature of DS is that it uses common format data files, which facilitates switching from one program to another. We applied DS to a three-dimensional ecological model by Dr. A. S. Kondrashov (one of the variables has a genetic meaning). Because of the rigid interface, the necessity of operating within DOS, and many strange conventions in the batch files, we did not enjoy working with DS very much. Nevertheless, we think that an interested user can learn DS quite easily. DS turned out to be useful for us and allowed us to discover some new features of the model. Dynamical Software is an interesting collection of programs for analyzing mathematical models and time series that may be helpful in many applications.
SOFTWARE
147
REVIEW
At the same time, it cannot be regarded as a modem interactive tool for PCs. The authors of DS compared the package with the well-known program PI-LASER in Nonlinear News (no. 6, 1989). It would be more appropriate to compare DS with INSITE (Proc. IEEE, vol. 75, no. 8, 1987), which also provides some nontrivial possibilities to study chaotic dynamics. The development of interactive programs for the analysis of bifurcations and chaos in nonlinear systems using a modem interface remains an unfinished task.
LaboratoT
ALEXANDER I. KHIBNIK of Mathematical Problems in Biology Research Computing Center Academy of Science of the USSR Pushchino, h4oscow region 142292 USSR
REVIEW
Dynamical Software I and II, for IBM PC or compatible with graphics and Microsoft or Ryan-McFarland FORTRAN compiler. Dynamical Systems, Post Office Box 35241, Tucson, Arizona 85740; (601) 825-1331. I, $250; II, $350; both $550.
Since 1963, when E. Lorenz described chaotic oscillations in a simple threedimensional model of thermoconvection, dynamical chaos and strange attractors have become extremely popular objects in mathematics and applied sciences. Many mathematicians, physicists, and researchers from other fields united their efforts in developing a theory of strange attractors. This theory produced an explosion of interest in computer simulation and forced many “pure” mathematicians to treat computers as a necessary and natural research tool. Numerical programs for nonlinear dynamics appearing in the last decade have, in turn, made these beautiful and highly developed theories accessible to many applied researchers. In the early 198Os, F. Takens and coworkers proposed a simple procedure to study dynamical chaos through experimental time series that does not assume the existence of a mathematical model of the considered process. The main idea is to introduce an appropriate time lag and to embed the original univariate time series into a two-, three-, or higher-dimensional space. The result is a trajectory in an m-dimensional space similar to a phase trajectory of a dynamical system, which then gives a graphical image of experimental chaotic oscillations in the form of an attractor comparable to known attractors or to the attractor in a relevant model. If some essential properties of the attractor (e.g., its fractal dimension) stabilize while we increase the embedding dimension, a finite-dimensional model of the process becomes possible. The package Dynamical Software (DS) is a toolkit for studying nonlinear dynamics and chaotic oscillations on an IBM PC or PC-compatible computer. DS is equally useful with a mathematical model or only experimental data. A model may be in the form of ordinary or delay differential equations or discrete maps. Using DS you can compute and draw the trajectories for any model and calculate Lyapunov exponents as well as smoothed power spectra for ODEs. DS has two procedures for bifurcation analysis of discrete maps. It can compute and plot a bifurcation diagram with which you can keep track of the evolution of the asymptotic state when a control parameter changes with a given step. One can also plot the rotation numbers against the control parameter. These procedures are the only methods of bifurcation analysis that DS supports. The bulk of the DS package is made up of programs for studying a computed trajectory or a trajectory produced from a univariate time series by means of the
MATHEMATICAL
BIOSCIENCES
98:145-147
OElsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010
(1990)
145 0025-5564/90/$03.50
146
SOFTWARE
REVIEW
embedding procedure. This part includes common numerical tools such as smoothing and interpolation, approximation of invariant measure by a histogram, statistics, and plotting a trajectory in different forms (two-dimensional and three-dimensional projections, color-valued velocity or an additional variable, and so on). The other procedures are more complex and time-consuming algorithms such as calculation of power spectra (periodogram, autocovariance function, and smoothed spectra), estimation of fractal dimension by correlation dimension, and computation of the maximal Lyapunov value from a time series. DS is distributed in two self-contained versions. As the authors write, DSI includes basic tools while DS II includes more advanced algorithms. The main features are simulation of mathematical models (I and II), phase portraits (I), discrete maps from time series (I), smoothing and interpolation (I), spectral analysis (II), fractal dimension (II), eigenvalues and Lyapunov exponents (II), transformations and statistics (II), and graphic utilities (I and II). The simple organization of DS enables the adding of new algorithms in the future. The DS manual is very detailed and contains a theoretical introduction to dynamical chaos that is illustrated by figures produced by DS, a description of the package, and many interesting examples, including the original experimental data from biology and medicine. Some of these examples are provided on the distribution disks on the form of tests and samples. Of the 20 distribution diskettes, probably half contain examples, graphics, or data sets. Unfortunately, many important details about the numerical algorithms used have been left out of the framework of the manual. Some interactive features and special graphic facilities are necessary because numerous simulations can be needed for a proper choice of parameters. DS supports many graphics adapters including CGA, EGA, and Hercules (but not VGA) and provides a convenient means for two- and three-dimensional plots. The weakest feature of the DS interface is the question-and-answer style it uses. You have to react to many prompts by typing various numbers that must be kept in mind. It is interesting that the same numerical codes result in different actions in different DS programs. Computation and plotting of results are usually done separately. Dynamical Software is only a software package, not an integrated environment, which is now becoming standard in the personal computer world. The DS user deals with the DOS operating system, files, a FORTRAN compiler, and linker. To make using DS more pleasant, the authors have provided some batch files and an on-line manual. An important useful feature of DS is that it uses common format data files, which facilitates switching from one program to another. We applied DS to a three-dimensional ecological model by Dr. A. S. Kondrashov (one of the variables has a genetic meaning). Because of the rigid interface, the necessity of operating within DOS, and many strange conventions in the batch files, we did not enjoy working with DS very much. Nevertheless, we think that an interested user can learn DS quite easily. DS turned out to be useful for us and allowed us to discover some new features of the model. Dynamical Software is an interesting collection of programs for analyzing mathematical models and time series that may be helpful in many applications.
SOFTWARE
147
REVIEW
At the same time, it cannot be regarded as a modem interactive tool for PCs. The authors of DS compared the package with the well-known program PI-LASER in Nonlinear News (no. 6, 1989). It would be more appropriate to compare DS with INSITE (Proc. IEEE, vol. 75, no. 8, 1987), which also provides some nontrivial possibilities to study chaotic dynamics. The development of interactive programs for the analysis of bifurcations and chaos in nonlinear systems using a modem interface remains an unfinished task.
LaboratoT
ALEXANDER I. KHIBNIK of Mathematical Problems in Biology Research Computing Center Academy of Science of the USSR Pushchino, h4oscow region 142292 USSR