Abstract
In finite precision implementations of chaotic maps all trajectories are eventually periodic. The goal of this brief is to develop methods for systematic study of effects of finite precision computations on dynamical behaviors of discrete maps and to carry out a study of the logistic map in this context. In particular, we are interested in finding all cycles when the logistic map is implemented in single and double precision and studying properties of these cycles including the size of the basin of attraction, and the maximum and average convergence times.
Highlights
W HEN chaotic maps are implemented using finite precision computations, the quantization causes dynamical degradation
An extra care has to be taken in verifying that a computer generated trajectory is not one of short cycles. These results indicate that double-precision implementations of the logistic map may cause serious disadvantages in chaos based applications like pseudo random number generators or chaos-based communication
Systematic methods to find all periodic solutions of finite precision implementations of chaotic maps have been proposed
Summary
W HEN chaotic maps are implemented using finite precision computations, the quantization causes dynamical degradation. The most common approach to solve this problem is based on computing trajectories for randomly or uniformly selected initial conditions In this approach one considers a large number of initial conditions and finds the corresponding steady states. We consider the logistic map [18] with 32-bit (single precision) and 64-bit (double precision) floating-point implementations As it will be show, the research problem to find all cycles is very challenging due to the size of the state space. A systematic study of dynamical behaviors of the logistic map with a = 3.9 and a = 4 implemented in the single and double precision floating-point formats using the computational formula fa(x) = a ∗ x ∗ (1 − x) is carried out
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