What chaos is and is not

We now get to a workable definition of chaos and clear up a few misconceptions. A recent definition was proposed by Cushing and colleagues in their book Chaos in Ecology.25 In their definition, which we will use ourselves, they combine elements of disorder, 'boundedness' and sensitivity to initial conditions all in one: 'a trajectory is chaotic if it is bounded in magnitude, is neither periodic nor approaches a periodic state, and is sensitive to initial conditions'. So, it is the sensitivity to initial conditions that provides a key clue to chaotic dynamics.

The first potential misconception is easily cleared up by pointing out that chaos is not only a property of mathematical models expressed in terms of difference equations. We introduced chaos through a simple difference equation, but models based on continuous changes can also exhibit chaos—indeed the two strange attractors in Figs. 6.5 and 6.6 were generated by models with continuous rates of change involving three dynamical variables. It turns out that chaos only occurs in simple differential equation systems involving three or more variables,20 but the possibilities for chaos get richer as we increase the number of variables.26 As mathematician Mark Kot27 noted, 'As soon as you move to three or more species, there are hundreds of ways to get chaos'. Second, while early researchers were taken aback by the complex dynamics predicted by simple sets of equations with no elements of chance involved (so-called deterministic equations), and many investigators continue to emphasize chaos as a primarily deterministic phenomenon, work has also been done to understand the role of small random elements (noise) in these chaotic systems.22 For example, small amounts of noise added to the dynamic can make something of a mess of bifurcation diagrams we described earlier, but the underlying bifurcations are still evident and the extreme sensitivity to initial conditions remains.28 Despite this, depending on one's specific definitions, noise may have the potential to turn non-chaotic systems intrinsically chaotic,29 thereby creating much more unpredictability than one would expect from the random elements alone.

The role of noise is currently under debate30-32 but it is clear that noise may do much more than provide a fuzzy cloud around a deterministic skeleton. We leave further consideration of the influence of such 'stochasticities', particularly in connection with cycling populations, until later in this chapter.

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