Modeling and Simulation of Complex Dynamical Systems

The six authors have written a detailed account of how students and teachers can use the Wolfram System Modeler to construct computer models of various systems. The systems are arranged in “virtual laboratories” covering such ideas as: projectile motion, a spring-damper system, rotational motion of bodies, a double pendulum, and a rotating swing, to name a few. 

 

The laboratories are developed with the equations of motion such as a system of differential equations. The authors show the reader how to use the Mathematica System Modeler to represent these equations and, if needed, any associated components for connectivity of the system, to run the model in Mathematica. The authors do a wonderful job of detailing the various parameters and showing the experimenter how to use the program to plot system variables over time or as a function of other variables. There are tables to organize the parameters and, quite frankly, the illustrations are better than good.

 

The entire book consists of just two chapters. Chapter one introduces the various systems to model and explore. Chapter two gives the experimenter hints and guides to creating the computer environment and run the models. It’s all lovely and picturesque. 

 

What’s not to like? Well, there are these:

 

From a computer software perspective, there is the cost. A look at the website shows the cost for the software to be as much at $7520 for an industry premier version, to $1970 for colleges as a personal license, to $526 for home or hobbyist users. Students can purchase a stripped-down version for $246 without Mathematica and no support. This is where an open-source program would likely be a better fit. I have been a fan of Octave and while I am not familiar with a toolbox for Octave that’s as visually impressive as Mathematica, I think much of what the System Modeler offers can be accomplished in Octave. 

 

The next quibble is the concept of a system model approach. By this I mean the use of building blocks to represent a system. Ordinarily, this approach is exactly what a student needs to understand the overall functions of a system and the various constituent components. However, that’s at a top-level design. These components in the book provide the input/output variables so that the actual operations inside the components are hidden. That might be useful but it is far better for a student or designer to know the detailed operations of the components. Blocks are fine, but knowing the internal mechanics is paramount to good research, design, and testing.

 

A third issue is the whole idea of modeling. Models have their place in understanding the world, designing a system, and sometimes even testing a system. There are, to be sure, many systems that cannot be tested or only tested limitedly. Weapon systems, for example, are not generally fully tested so testers use models to develop the performance envelopes of, say, missiles. The Navy, for example, does not test fire torpedoes at ships. In that case, a system model is fine and might be all that can be done. What’s not so fine is when students are taught, as this text implies, that the model is the end of the system design.

 

Computer models need to be seen for what they are: a limited look at a complex design. I am afraid this book (and many, many others like it) promote the computer model as the end. The model is the start. It tells the experimenter what might happen, what variables may be important to stress, and how the system might function at its limits. This book implies the model is it; I beg to differ. I wish the authors had noted the many limits of models and their usefulness. 

 

Some models are physical manifestations of the object of study. For example, Lego makes a model of the Bugatti Chiron that is a close replica of the actual sports car. The model is to scale, it has a twelve-cylinder engine, a differential gearbox, steering, and even a shifter that “works.” I built it and it’s on a shelf in my study. It’s not the car but it looks like it. Is this model useful? Maybe. But its utility is quite limited. The models in this book are limited, too, but in a different manner.

 

Other issues, and minor ones at that, are: there is no index, effects of round-off errors with non-linear models are not discussed, the gyroscope example is difficult to follow; and some of the tables and figures are hard to read (see Tables 1.29 and 1.38 as examples).

 

On the whole, the authors went to great effort to produce beautiful figures, with color no less, to aid the reader. The tables are laid out logically, and the discussion for each model is, for the most part, straightforward. I wish I could recommend the book but frankly, I think an experimenter would be better off with a book on physics, Octave, and writing his own code.

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David S. Mazel is a practicing engineer in Washington, DC. He welcomes your thoughts and feedback. He can be reached at mazeld at gmail dot com.