Unscrambling the Concept of Chaos through Thick and Thin: Reply

Abstract

Melese and Transue rightly distinguish two types of chaos: that in which the scrambled set has positive measure (observable chaos); and that when it has zero measure (unobservable chaos). In a series of studies that began with the reference they cite, Wayne Shafer and I have explored the existence of observable chaos for a Keynesian business cycle model [Day and Shafer, 1985a,b; Day, 1985]. On the basis of this work it is clear that observable chaos can exist and be robust for a continuum of parameter values for a large class of difference equations that arise quite naturally in economics. The functional forms for this class involve piece-wise smooth functions-in effect, switching regimes. Readers concerned with these issues may also be interested to learn of extensive computational studies, carried out by Pianigiani, of the classic quadratic difference equation that I used as a special case of a more general example and to which Melese and Transue confine their attention. These simulations, which were shown to me in Siena and which augment similar studies by James Yorke (both of these scholars are among the pioneers in the field), show that throughout the range of values 3.57 < m < 4.00, behavior switches from stable, relatively loworder cycles (with thin chaos) to very high-order cycles or thick chaos. The high-order cycles appear to be nonperiodic for finite series shorter than their periodicity. The latter could be in the hundreds, thousands, millions, or higher! One should also be reminded that Lorenz [1963], in one of his seminal papers, provided evidence that the asymptotic behavior of solutions changed erratically within what is now known to be the chaos zone. Putting all this together, we see that (1) thick chaos can be robust in important classes of models; (2) thin chaos may frequently be associated with stable, but high-order cycles that are practically indistinguishable from chaos; and (3) the long-run behavior of nonlinear difference equations typically switches frequently and sometimes erratically with parameter shifts. In a nutshell, complex, essentially unpredictable behavior is a structurally stable feature of nonlinear difference equations through