Reconstruction of one-dimensional chaotic maps from sequences of probability density functions

Nonlinear Dynamics, Feb 2015

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.

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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 - 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)


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Xiaokai Nie, Daniel Coca. Reconstruction of one-dimensional chaotic maps from sequences of probability density functions, Nonlinear Dynamics, 2015, pp. 1373-1390, Volume 80, Issue 3, DOI: 10.1007/s11071-015-1949-9