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nonlinear behavior cannot be modeled adequately with linear models. Although once this behavior is
recognized through the use of a nonlinear model, the resulting steady-state system (a stable oscillator)
can be modeled for many purposes using its linear (although physically unrealizable) equivalent.
When using differential equations to model a system, we must be particularly careful in our choice of
units of measure and in how we group parameters. For example, in the Van der Pol equation above, we
can choose an alternative scale for y such that y 0 is 1 and a time scale such that ω is 1. In this case we have

(
2
–
z′′ = −z + e 1 z )z′
where z = y/y 0 , τ = ω t, and ε = α/ω. This scaling provides a dimensionless equation as well as providing
an indication of the system’s natural time scale, the scale of y, and an indication that the size of α relative
to ω is important. In general, we find that writing equations in dimensionless form provides three
advantages. First, as mentioned above, the scaling factors provide a sense of time scale and magnitude of
various features of the system’s dynamic behavior. Second, the dimensionless parameters that result from
combining physical system parameters provide a way of characterizing the behavior of a wide range of
systems in terms of parameters that are easily mapped. Third, by putting the equation into dimensionless
form, we are able to categorize it and recognize similar equations in different domains.

References
1. Hubka, V. and Eder, W.E., Engineering Design—General Procedural Model of Engineering Design,
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10. Karnopp, D. and Rosenberg, R., System Dynamics: A Unified Approach, Wiley, New York, 1975.
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Dearborn MI, 1994.
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14. Lonergan, B., Method in Theology, University of Toronto, Toronto, 1990.
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