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Time permeates our existence, saturating our very being. Nevertheless, few things are as difficult to comprehend and as hard to model mathematically. Although the invention of the calculus thrust dynamics dramatically forward, only the simplest cases proved solvable. In many important cases, directly solving systems of differential equations is intractable. Alternative techniques are required.
Modern methods investigate the qualitative properties of entire families or flows of trajectories rather than solving for a single trajectory determined by particular parameters and initial conditions. Fundamental to this alternative approach is the identification of initial conditions and parameter values at which structural qualities change, thereby revealing much about the underlying processes governing motion.
These qualitative techniques, which involve the topological properties of the flow, have been applied to several scientific disciplines. Their reception, however, has been uneven. They have had successes in the study of the motion of fluids, for example, but have met less enthusiasm, if not open hostility, among economists.
This chapter documents and explains the differing reception of the mathematics of qualitative dynamics in economics as opposed to hydrodynamics. The first section outlines the mathematics involved, especially the notion of structural instability and its role in generating complex motion. The second describes application of these techniques to two problems in fluid mechanics. The third examines attempts to employ similar methods in economics, arguing that they have not brought great success there. Finally, the fourth section suggests that the contrast between acceptance of the same mathematics in these two fields arises first from their different empirical and evidentiary foundations and second from the distinct cultural and metaphorical background of each.
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