1,361 research outputs found
Forbidden ordinal patterns in higher dimensional dynamics
Forbidden ordinal patterns are ordinal patterns (or `rank blocks') that
cannot appear in the orbits generated by a map taking values on a linearly
ordered space, in which case we say that the map has forbidden patterns. Once a
map has a forbidden pattern of a given length , it has forbidden
patterns of any length and their number grows superexponentially
with . Using recent results on topological permutation entropy, we study in
this paper the existence and some basic properties of forbidden ordinal
patterns for self maps on n-dimensional intervals. Our most applicable
conclusion is that expansive interval maps with finite topological entropy have
necessarily forbidden patterns, although we conjecture that this is also the
case under more general conditions. The theoretical results are nicely
illustrated for n=2 both using the naive counting estimator for forbidden
patterns and Chao's estimator for the number of classes in a population. The
robustness of forbidden ordinal patterns against observational white noise is
also illustrated.Comment: 19 pages, 6 figure
Permutation Complexity via Duality between Values and Orderings
We study the permutation complexity of finite-state stationary stochastic
processes based on a duality between values and orderings between values.
First, we establish a duality between the set of all words of a fixed length
and the set of all permutations of the same length. Second, on this basis, we
give an elementary alternative proof of the equality between the permutation
entropy rate and the entropy rate for a finite-state stationary stochastic
processes first proved in [Amigo, J.M., Kennel, M. B., Kocarev, L., 2005.
Physica D 210, 77-95]. Third, we show that further information on the
relationship between the structure of values and the structure of orderings for
finite-state stationary stochastic processes beyond the entropy rate can be
obtained from the established duality. In particular, we prove that the
permutation excess entropy is equal to the excess entropy, which is a measure
of global correlation present in a stationary stochastic process, for
finite-state stationary ergodic Markov processes.Comment: 26 page
Entropy increase in switching systems
The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them “macroscopic” and “microscopic” complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondo’s paradox
Composition law of cardinal order permutations
In this paper the theorems that determine composition laws for both cardinal
ordering permutations and their inverses are proven. So, the relative positions
of points in a hs-periodic orbit become completely known as well as in which
order those points are visited. No matter how a hs-periodic orbit emerges, be
it through a period doubling cascade (s=2^n) of the h-periodic orbit, or as a
primary window (like the saddle-node bifurcation cascade with h=2^n), or as a
secondary window (the birth of a periodic window inside the h-periodic
one). Certainly, period doubling cascade orbits are particular cases with h=2
and s=2^n. Both composition laws are also shown in algorithmic way for their
easy use
Energy and Personality: A Bridge between Physics and Psychology
[EN] The objective of this paper is to present a mathematical formalism that states a bridge
between physics and psychology, concretely between analytical dynamics and personality theory,
in order to open new insights in this theory. In this formalism, energy plays a central role. First,
the short-term personality dynamics can be measured by the General Factor of Personality (GFP)
response to an arbitrary stimulus. This GFP dynamical response is modeled by a stimulus¿response
model: an integro-differential equation. The bridge between physics and psychology appears when
the stimulus¿response model can be formulated as a linear second order differential equation and,
subsequently, reformulated as a Newtonian equation. This bridge is strengthened when the Newtonian
equation is derived from a minimum action principle, obtaining the current Lagrangian
and Hamiltonian functions. However, the Hamiltonian function is non-conserved energy. Then,
some changes lead to a conserved Hamiltonian function: Ermakov¿Lewis energy. This energy is
presented, as well as the GFP dynamical response that can be derived from it. An application case
is also presented: an experimental design in which 28 individuals consumed 26.51 g of alcohol.
This experiment provides an ordinal scale for the Ermakov¿Lewis energy that predicts the effect of
a single dose of alcohol.Caselles, A.; Micó, JC.; Amigó, S. (2021). Energy and Personality: A Bridge between Physics and Psychology. Mathematics. 9(12):1-20. https://doi.org/10.3390/math9121339S12091
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