Clarifying chaos II: Bernoulli chaos, zero Lapunov exponents and strange attractors

In this tutorial we continue the program initiated in "Clarifying Chaos: Examples and Counter Examples" by presenting examples that answer questions in five areas: Area 1. The Horseshoe/Bilateral Shifts/Bernoulli Systems Since the bilateral shift (which may also be called a Bernoulli shift) plays such an important role in some definitions of chaos we show that it is possible to construct a differential equation for an electronic circuit whose time-one map,(1) is exactly a bilateral shift, in particular the bakers transformation and the cat map [Arnold & Avez, 1989]. We insist on being able to build a circuit in order to be sure that our example is not just a mathematical abstraction. Also, in this set of examples we show that we may construct chaotic maps of any desired level of complexity. Area 2. Zero Lyapunov Exponents Since the existence of positive Lyapunov exponents is so often used as a definition of chaos we answer the question: Are there systems with zero Lyapunov exponents which are not considered chaotic by this definition, which have outputs which are more complex that some chaotic systems? The answer is yes, and for these systems, called skew translations and compound skew translations [Cornfeld et al, 1982], all the eigenvalues are 1. Further, the skew translation may be linear, having only additions (no multiplication's). Skew translations exist in any number of dimensions and can be realized as the time-one maps of an electronic circuit. Skew translations can have sensitive dependence on initial conditions and zero autocorrelations. The significance of this example is that the Lyapunov exponent is less a measure of the level of complexity than one first imagined since a higher level of complexity can be obtained from a lower exponent. Area 3. Nonchaotic Strange Attractors This phenomenon is reported in [Grebogi et at, 1984] and further developed by other researchers. Of note in this regard is the work of Ding et al. [1989] where the place of this phenomenon within nonlinear dynamics is discussed. We show here that the origin of this phenomenon is found in dynamical systems having orbits with low correlations regardless of their Lyapunov exponents. We present examples of skew translations having zero autocorrelations and zero Lyapunov exponents that can be used to generate nonchaotic strange attractors. Further, we show that only minimal level of complexity is needed to obtain nonchaotic strange attractors by using a group rotation to produce one. The inverse of this idea is the chaotic nonstrange attractor which is also presented. Area 4. Nonlinearity Since nonlinearities are usually considered a key ingredient of chaotic dynamical systems we present examples to show that there are at least four distinct types of nonlinearities in ODEs leading to varying levels of chaos. All example ODEs have closed-form solutions in terms of elementary functions and thus give us direct insight into how the type of nonlinearity appears in the ODE and is manifested in its solution. Area 5. Relationship of Dissipation, Noninvertibility, Nonorientibility and Chaos There are many misconceptions about how these properties, especially dissipation, may contribute to chaos. We show that these properties are independent of chaos. The overriding conclusion of this set of examples is that what we have traditionally called chaos is so varied in its level of complexity that it is almost a meaningless term when used by itself. In particular, the term "level of complexity" must be appealed to so often in order to clarify the varying degrees of chaos that the two terms "chaos" and "level of complexity" seem inseparable in any practical discussion of chaos. The key issue that gives rise to this confusion about the level of complexity of a chaotic dynamical system is its long- and shortterm predictability. Chaotic dynamical systems may be quite predictable over very long but finite time scales, but unpredictable ln infinite time. The need to consider system behavior over long, finite time scales is a practical matter and leads to the conclusion that the study of chaos must be concerned with both asymptotic and long, but finite, time dynamics.