Estimation of the control parameter from symbolic sequences: Unimodal maps with variable critical point

The work described in this paper can be interpreted as an application of the order patterns of symbolic dynamics when dealing with unimodal maps. Specifically, it is shown how Gray codes can be used to estimate the probability distribution functions (PDFs) of the order patterns of parametric unimodal maps. Furthermore, these PDFs depend on the value of the parameter, what eventually provides a handle to estimate the parameter value from symbolic sequences (in form of Gray codes), even when the critical point depends on the parameter.Comment: 10 pages, 14 figure

Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps

In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works

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

Abstract: The symbolic time series generated by a unimodal chaotic map starting from any initial condition creates a binary sequence that contains information about the initial condition. A binary sequence of a given length generated this way has a one-to-one correspondence with a given range of the input signal. This can be used to construct analogue to digital converters (ADC). However, in actual circuit realizations, component imperfections and ambient noise result in deviations in the map function from the ideal, which, in turn, can cause significant error in signal measurement. In this paper, we propose the ways of circumventing these problems through an algorithmic procedure that takes into account the non-idealities. The most common form of non-ideality--reduction in the height of the map function--alters the partitions that correspond to each symbolic sequence. We show that it is possible to define the partitions correctly if the height of the map function is known. We also propose a method to estimate this height from the symbolic sequence obtained. We demonstrate the efficacy of the proposed algorithm with simulation as well as experiment. With this development, practical ADCs utilizing chaotic dynamics may become reality

Estimation of Input Variable as Initial Condition of a Chaos Based Analogue to Digital Converter

A realization of an analogue-to-digital converter(ADC) with improved conversion accuracy,using the chaotic behaviour of the tent map,is presented. In this approach, the analogue input signal to be measured, termed as the initial condition is applied to a chaotic map, and the symbolic dynamics resulting from the map evolution, is used to determine the initial condition in digital form. The unimodal piecewise linear tent map (TM) has been used for this purpose, because of its property of generating uniform distribution of points and robust chaos. Through electronic implementation of the TMit is practically impossible to produce an ‘ideal’ TM behaviour with parameter values in the full range [0,1]. Due to component imprecision and various other factors, a non-ideal map with reduced height is observed. For such a map, converting the equivalent symbolic trajectory generated by TM iterations return erroneous results as the partitioning of the phase space embodied in the finite symbolic dynamics no longer has unique correspondence with the initial condition. Two algorithmic solutions have been proposed to minimise the errors associated with a practical system. For one, it has been established that for a reduced-height map the partitioning will not remain of equal size. Considering that the height of the tent map used for this purpose is known from an independent but related research, a technique of partitioning the state space unevenly, depending on the map height has been proposed and has been shown that if the correct partitioning is used, the resulting symbolic dynamics again map uniquely to the initial condition. Alternatively, it has been shown that the degree of deviation of the iterate values can be determined based on the parameter value, which in turn can be adjusted for depending on the symbolic sequence generated by the initial condition to determine the correct decimal equivalent values. The both the approaches proved to be highly effective in obtaining a digital outcome corresponding to the initial condition using 8 symbolic iterations of the map in hardware domain, with the second approach outperforming the first in terms of accuracy, while the first method can easily be pipelined alongside generating the iterates and thus improve the speed. This development is promising because, in contrast to the commercially available ADCs, it places lower demand on the hardware resource and can be effectively implemented to give a real-time operation

Estimation of Input Variable as Initial Condition of a Chaos Based Analogue to Digital Converter

A realization of an analogue-to-digital converter(ADC) with improved conversion accuracy,using the chaotic behaviour of the tent map,is presented. In this approach, the analogue input signal to be measured, termed as the initial condition is applied to a chaotic map, and the symbolic dynamics resulting from the map evolution, is used to determine the initial condition in digital form. The unimodal piecewise linear tent map (TM) has been used for this purpose, because of its property of generating uniform distribution of points and robust chaos. Through electronic implementation of the TMit is practically impossible to produce an ‘ideal’ TM behaviour with parameter values in the full range [0,1]. Due to component imprecision and various other factors, a non-ideal map with reduced height is observed. For such a map, converting the equivalent symbolic trajectory generated by TM iterations return erroneous results as the partitioning of the phase space embodied in the finite symbolic dynamics no longer has unique correspondence with the initial condition. Two algorithmic solutions have been proposed to minimise the errors associated with a practical system. For one, it has been established that for a reduced-height map the partitioning will not remain of equal size. Considering that the height of the tent map used for this purpose is known from an independent but related research, a technique of partitioning the state space unevenly, depending on the map height has been proposed and has been shown that if the correct partitioning is used, the resulting symbolic dynamics again map uniquely to the initial condition. Alternatively, it has been shown that the degree of deviation of the iterate values can be determined based on the parameter value, which in turn can be adjusted for depending on the symbolic sequence generated by the initial condition to determine the correct decimal equivalent values. The both the approaches proved to be highly effective in obtaining a digital outcome corresponding to the initial condition using 8 symbolic iterations of the map in hardware domain, with the second approach outperforming the first in terms of accuracy, while the first method can easily be pipelined alongside generating the iterates and thus improve the speed. This development is promising because, in contrast to the commercially available ADCs, it places lower demand on the hardware resource and can be effectively implemented to give a real-time operation

On the universal sequence generated by a class of unimodal functions

AbstractThe universality of the Metropolis, Stein, and Stein (MSS). sequence (J. Combin. Theory 15 (1973), 25–44) is established for a wide class of unimodal functions. The standard value of an LR-sequence is defined and a computational formula for it is established. An order on all finite LR-sequences is defined. It is shown that this order is equivalent to the order of Collet and Eckman (CE) (“Iterated Maps on the Interval as Dynamcal Systems,” Birkhauser, Boston, 1980), Louck and Metropolis (“Symbolic Dynamics of Trapezoidal Maps,” Reidel-Kluwer, Hingham, Ma, 1986) and Beyer, Mauldin, and Stein (BMS), (J. Math.Anal. Appl. 115 (1986), 305–362). The contiguity of harmonics is proved for any finite LR-sequence: Finally using an important result of BMS, it is shown that a pattern is legal if and only if it is a pattern associated with a positive solution λ of , the sequence of equations [λf]k(yo)=y0 (k= 1, 2,…)