Forbidden patterns and shift systems

Abstract

The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms.

Quantifying stochasticity in the dynamics of delay-coupled semiconductor lasers via forbidden patterns

“…Identifying the existence of such forbidden patterns in our time series should tell us whether they have an underlying deterministic behaviour. This technique has been used to characterize the level (or lack) of stochasticity in logistic maps (Amigó et al 2006), shift systems (Amigó et al 2008), and financial time series (Zanin 2008;Zunino et al 2009). Here we report on what is, as far as we know, the first application of this method to experimentally generated time series.…”

Quantifying the complexity of the delayed logistic map

“…As discussed in §2, Amigó and colleagues [42,45,46,48] have shown that, for deterministic maps, not all ordinal patterns can be materialized in the orbits, not even if the dynamics is chaotic, which makes these patterns 'forbidden'. For example, in time series generated by the undelayed logistic map with r = 4, if we consider patterns of length D = 3, the ordinal pattern (2 1 0) is forbidden.…”

“…The patterns that are not observed in a time series are referred to as 'missing patterns' and will be discussed in §3. It has been shown that piecewise monotone onedimensional maps always have 'forbidden patterns', which are permutations that do not appear in any trajectory obtained by iterating the map, starting from any point in the interval [45,46]. Basic or minimal forbidden patterns, which are those for which any proper consecutive subpattern is allowed, have been studied for various maps [47].…”

An Algorithmic Approach for Signal Measurement Using Symbolic Dynamics of Tent Map

“…Firstly, when the height of the map is less than 1, the map dynamics eventually gets confined within a range of the state space. Therefore, some sequences corresponding to the points that are not visited by the dynamics will not be appearing as well in the symbolic dynamics; those sequences are classified as forbidden sequences [10]. We have shown that, even though there are forbidden sequences, information regarding the points outside the bounding region can still be realised using the symbolic dynamics produced within the boundary.…”