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# 9.2: Characteristics of Chaos

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It is helpful to realize that there are two dynamical processes always going on in any kind of chaotic systems: stretching and folding [33]. Any chaotic system has a dynamical mechanism to stretch, and then fold, its phase space, like kneading pastry dough (Figure $$\PageIndex{1}$$). Imagine that you are keeping track of the location of a speciﬁc grain of ﬂour in the dough while a pastry chef kneads the dough for a long period of time. Stretching the dough magniﬁes the tiny differences in position at microscopic scales to a larger, visible one, while folding the dough always keeps its extension within a ﬁnite, conﬁned size. Note that folding is the primary source of nonlinearity that makes long-term predictions so hard—if the chef were simply stretching the dough all the time (which would look more like making ramen), you would still have a pretty good idea about where your favorite grain of ﬂour would be after the stretching was completed. This stretching-and-folding view allows us to make another interpretation of chaos:

Stretching and Folding Phase Space

Chaos can be understood as a dynamical process in which microscopic information hidden in the details of a system’s state is dug out and expanded to a macroscopically visible scale (stretching), while the macroscopic information visible in the current system’s state is continuously discarded (folding).

This kind of information ﬂow-based explanation of chaos is quite helpful in understanding the essence of chaos from a multiscale perspective. This is particularly clear when you consider the saw map discussed in the previous exercise:

$x_{t}= \text{fractional part of } 2x_{t-1} \label{9.1}$

If you know binary notations of real numbers, it should be obvious that this iterative map is simply shifting the bit string in $$x$$ always to the left, while forgetting the bits that came before the decimal point. And yet, such a simple arithmetic operation can still create chaos, if the initial condition is an irrational number (Figure $$\PageIndex{2}$$)! This is because an irrational number contains an inﬁnite length of digits and chaos continuously digs them out to produce a ﬂuctuating behavior at a visible scale.

Exercise $$\PageIndex{1}$$

The saw map can also show chaos even from a rational initial condition, if its behavior is manually simulated by hand on a cobweb plot. Explain why.