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-toone 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 ways of circumventing these problems through an algorithmic procedure that takes into account the non-idealities. The most common form of nonideality-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.
Many physical situations involve chaotic systems implemented in hardware. Among them onedimensional piecewise linear maps are popular candidates for such applications because of their property of generating robust chaos. In physical implementations, the control parameter of these maps may deviate from its ideal value due to hardware imprecision. Since the dynamics of a chaotic map is completely defined by its control parameter, one needs to know the value of the parameter in a hardware realisation. In this paper, we show that it is possible to determine the parameter, through the realisation of the unstable fixed point of the map, by utilising noise that is always present in the system. We present this in the form of an algorithm and demonstrate its efficacy through simulated results. We also determine the bounds on the signal-to-noise ratio required for successful parameter estimation. The proposed approach is expected to be beneficial to the existing noise reduction techniques and time series recovery algorithms that require a reasonably accurate knowledge of the map.
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