Ordinal Patterns

In this chapter we take a close look to the order relations and their consequences, mostly for dynamical systems defined by self-maps of one-dimensional intervals. More general situations will be considered and studied in detail in the following chapters. Order has some interesting consequences in discrete-time dynamical systems. Just as one can derive sequences of symbol patterns from such a dynamic via coarse graining of the phase space, so it is also straightforward to obtain sequences of ordinal patterns if the phase space is linearly ordered. As we learnt in Sect. 1.2, not all ordinal patterns can be materialized by the orbits of a given dynamic under some mild mathematical assumptions. Furthermore, if an ordinal pattern of a given length is “forbidden,” i.e., cannot occur, its absence pervades all longer patterns in form of more missing ordinal patterns. This cascade of outgrowth forbidden patterns grows super-exponentially (in fact, factorially) with the length, all its patterns sharing a common structure. Of course, forbidden and admissible ordinal patterns can be viewed as permutations; in combinatorial parlance, the admissible patterns are (the inverses of) those permutations avoiding the so-called forbidden root patterns in consecutive positions (see Sect. 3.4.2 for details). Let us mention that permutations avoiding general or consecutive patterns is a popular topic in combinatorics (see, e.g., [25, 74, 75]).