Reconstruction of one-dimensional chaotic maps from sequences of probability density functions
Reconstruction of one-dimensional chaotic maps from sequences of probability density functions
Xiaokai Nie
Daniel Coca
In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius-Perron problem, that is, to reconstruct an unknown one-dimensional chaotic transformation, based on a temporal sequence of probability density functions generated by the transformation. Numerical examples are used to demonstrate the applicability of the proposed approach and evaluate its robustness with respect to constantly applied stochastic perturbations.
Chaotic maps; Inverse Frobenius-Perron problem; Nonlinear systems; Probability density functions
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physics and economics [1], which generate density of
states. Examples include particle formation in emulsion
polymerization [2], papermaking systems [3], bursty
packet traffic in computer networks [4,5], cellular
uplink load in WCDMA systems [6]. A major
challenge is that of inferring the chaotic map that describes
the evolution of the unknown chaotic system, solely
based on experimental observations.
Starting with seminal papers of Farmer and
Sidorovich [7], Casadgli [8] and Abarbanel et al.
[9], the problem of inferring dynamical models of
chaotic systems directly from time series data has been
addressed by many authors using neural networks [10],
polynomial [11] or wavelet models [12].
In many practical applications, it is more convenient
to observe experimentally the evolution of the
probability density functions, instead of individual point
trajectories, generated by such systems. For example,
the particle image velocimetry (PIV) method of flow
visualization [13] allows identifying individual tracer
particles in each image, but not to track these between
images. In biology, flow cytometry is routinely used to
measure the expression of membrane proteins of
individual cells in a population [14]. However, it is
impossible to track the cells between subsequent analyses.
The problem of inferring an unknown chaotic map
given the invariant density generated by the map
is known as the inverse FrobeniusPerron problem
(IFPP). This inverse problem has been investigated by
Friedman and Boyarsky [15] who treated the inverse
problem for a very restrictive class of piecewise
constant density functions, using graph-theoretical
methods. Ershov and Malinetskii [16] proposed a
numerical algorithm for constructing a one-dimensional
unimodal transformation which has a given invariant
density. The results were generalized in Gra and Boyarsky
[17], who introduced a matrix method for constructing
a 3-band transformation such that an arbitrary given
piecewise constant density is invariant under the
transformation. Diakonos and Schmelcher [18] considered
the inverse problem for a class of symmetric maps that
have invariant symmetric Beta density functions. For
the given symmetry constraints, they show that this
problem has a unique solution. A generalization of this
approach, which deals with a broader class of
continuous unimodal maps for which each branch of the map
covers the complete interval and considers
asymmetric beta density functions, is proposed in [19]. Huang
presented approaches to constructing smooth chaotic
transformation with closed form [20,21] and
multibranches complete chaotic map [22] given invariant
densities. Boyarsky and Gra [23] studied the problem
of representing the dynamics of chaotic maps, which
is irreversible by a reversible deterministic process.
Baranovsky and Daems [24] considered the problem of
synthesizing one-dimensional piecewise linear Markov
maps with prescribed autocorrelation function. The
desired invariant density is then obtained by
performing a suitable coordinate transformation. An
alternative stochastic optimization approach is proposed in
[25] to synthesize smooth unimodal maps with given
invariant density and autocorrelation function. An
analytical approach to solving the IFPP for two specific
types of one-dimensional symmetric maps, given an
analytic form of the invariant density, was introduced in
[26]. A method for constructing chaotic maps with
arbitrary piecewise constant invariant densities and
arbitrary mixing properties using positive matrix theory
was proposed in [5]. The approach has been exploited
to synthesize dynamical systems with desired
characteristics, i.e. Lyapunov exponent and mixing properties
that share the same invariant density [27] and to analyse
and design the communication networks based on
TCPlike congestion control mechanisms [28]. An extension
of this work to randomly switched chaotic maps is
studied in [29]. It is also shown how the method can be
extended to higher dimensions and how the approach
can be used to encode images. In [30], the inverse
problem (...truncated)