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1 Flows vs. Maps     continued...

As before, in order to obtain chaos for a map the equations (6) must contain at least one nonlinear term. However, for a map chaos can arise even for a one-dimensional system [3], which should be contrasted with the corresponding criterion for a flow. This is a very important point since, in order to study chaotic effects one need only consider one- or two-dimensional maps, whereas for a flow at least three dimensions is mandatory. Moreover, for a given dimensionality n, the map equations (6) are much easier to use than the flow equations (3). For a mapping, we study the dynamics by simply iterating, whereas for a flow we must solve a system of differential equations, which in general is nontrivial. Thus, for the remainder of this chapter we will focus on chaos in several simple one- and two-dimensional maps.