84 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
Forbidden patterns and shift systems
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.Comment: 21 pages, expanded Section 5 and corrected Propositions 3 and
Entropy Monotonicity and Superstable Cycles for the Quadratic Family Revisited
The main result of this paper is a proof using real analysis of the monotonicity of the
topological entropy for the family of quadratic maps, sometimes called Milnor’s Monotonicity
Conjecture. In contrast, the existing proofs rely in one way or another on complex analysis.
Our proof is based on tools and algorithms previously developed by the authors and collaborators to
compute the topological entropy of multimodal maps. Specifically, we use the number of transverse
intersections of the map iterations with the so-called critical line. The approach is technically simple
and geometrical. The same approach is also used to briefly revisit the superstable cycles of the
quadratic maps, since both topics are closely related
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
Internet congestion control: From stochastic to dynamical models
Since its inception, control of data congestion on the Internet has been based on stochas tic models. One of the first such models was Random Early Detection. Later, this model
was reformulated as a dynamical system, with the average queue sizes at a router’s
buffer being the states. Recently, the dynamical model has been generalized to improve
global stability. In this paper we review the original stochastic model and both nonlin ear models of Random Early Detection with a two-fold objective: (i) illustrate how a
random model can be “smoothed out” to a deterministic one through data aggregation
and (ii) how this translation can shed light into complex processes such as the Internet
data traffic. Furthermore, this paper contains new materials concerning the occurrence
of chaos, bifurcation diagrams, Lyapunov exponents and global stability robustness with
respect to control parameters. The results reviewed and reported here are expected to
help design an active queue management algorithm in real conditions, that is, when sys tem parameters such as the number of users and the round-trip time of the data packets
change over time. The topic also illustrates the much-needed synergy of a theoretical
approach, practical intuition and numerical simulations in engineerin
Generalized TCP-RED dynamical model for Internet congestion control
Adaptive management of traffic congestion in the Internet is a complex problem that can
gain useful insights from a dynamical approach. In this paper we propose and analyze
a one-dimensional, discrete-time nonlinear model for Internet congestion control at the
routers. Specifically, the states correspond to the average queue sizes of the incoming
data packets and the dynamical core consists of a monotone or unimodal mapping with a
unique fixed point. This model generalizes a previous one in that additional control param eters are introduced via the data packet drop probability with the objective of enhancing
stability. To make the analysis more challenging, the original model was shown to exhibit
the usual features of low-dimensional chaos with respect to several system and control pa rameters, e.g., positive Lyapunov exponents and Feigenbaum-like bifurcation diagrams. We
concentrate first on the theoretical aspects that may promote the unique stationary state
of the system to a global attractor, which in our case amounts to global stability. In a sec ond step, those theoretical results are translated into stability domains for robust setting of
the new control parameters in practical applications. Numerical simulations confirm that
the new parameters make it possible to extend the stability domains, in comparison with
previous results. Therefore, the present work may lead to an adaptive congestion control
algorithm with a more stable performance than other algorithms currently in use
New RED-type TCP-AQM algorithms based on beta distribution drop functions
In recent years, Active Queue Management (AQM) mechanisms to improve the
performance of TCP/IP networks have acquired a relevant role. In this paper we
present a simple and robust RED-type algorithm together with a couple of
dynamical variants with the ability to adapt to the specific characteristics of
different network environments, as well as to the user needs. We first present
a basic version called Beta RED (BetaRED), where the user is free to adjust the
parameters according to the network conditions. The aim is to make the
parameter setting easy and intuitive so that a good performance is obtained
over a wide range of parameters. Secondly, BetaRED is used as a framework to
design two dynamic algorithms, which we will call Adaptive Beta RED (ABetaRED)
and Dynamic Beta RED (DBetaRED). In those new algorithms certain parameters are
dynamically adjusted so that the queue length remains stable around a
predetermined reference value and according to changing network traffic
conditions. Finally, we present a battery of simulations using the Network
Simulator 3 (ns-3) software with a two-fold objective: to guide the user on how
to adjust the parameters of the BetaRED mechanism, and to show a performance
comparison of ABetaRED and DBetaRED with other representative algorithms that
pursue a similar objective
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