Stochastic Implementation and Analysis of Dynamical Systems Similar to the Logistic Map
Abstract : Stochastic computing (SC) is a digital computation approach that operates on random bit streams to perform complex tasks with much smaller hardware footprints compared with conventional binary radix approaches. SC works based on the assumption that input bit streams are independent random sequences of 1s and 0s. Previous SC efforts have avoided implementing functions that have feedback, because doing so has the potential for creating highly correlated inputs. We propose a number of solutions to overcome the challenges of implementing feedback in stochastic logic. We use a family of dynamical system functions that are similar to the well-known logistic map x ? µx(1- x) as case studies. We show that complex behaviors, such as period doubling and chaos, do indeed occur in digital logic with only a few gates operating on a few 0s and 1s. Our energy consumption is between 21% and 31% of the conventional binary approach. In order to verify our design methodology, we have measured the mean switching rate between the basins of attraction of two coexisting fixed points and the peak width of the steady-state distribution of the output using a logistic-map-like function as an example. Theoretical results match well with our numerical experiments.
? The existence of a network relationship among all possible states of a given chaotic map in the digital domain was often ignored, instead having focus on statistics along the orbit (path) on the network.
? Each weakly connected component has one and only one self-loop (an edge connecting a node to itself) or cycle (a sequence of nodes starting and ending at the same node such that, for every two consecutive nodes in the cycle, there exists an edge directed from the former node to the latter One.)
? As above enhancing methods can be considered to make the dynamical properties of an existing chaotic map become more complex, SMN can also be used to evaluate its dynamical complexity.
? The impact of increasing the computation resolution W on the hardware performance metrics of the different implementations.
? The implementation of a chaotic system in a digital device is always an inevitable problem withholding its real applications.
? The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence.
? The bit sequence generated is then applied to the problem of image encryption, and the resulting encrypted image is analysed for security using methods like histogram analysis, correlation, and information entropy.
• In, a set of objective metrics were proposed to measure the degree of dynamics degradation of piecewise-linear chaotic maps.
• An analytical framework was proposed for recurrence network analysis of chaotic time series.
• A large number of PRNG have been proposed based on various chaotic maps and their variants, e.g., the Logistic map, the Tent map, the Sawtooth map, the Renyi chaotic map, and the Cat map.
• To counteract dynamics degradation, many methods have been proposed, for example adopting higher precision, perturbing chaotic states, perturbing control parameters, and cascading multiple chaotic maps, switching multiple chaotic maps and feedback control.
? It can be seen that the relative performance of the different circuits does not change significantly when we use a higher resolution.
? It has been used in applications, such as low-density parity-check (LDPC) encoding, filter design, and image processing.
? The correlation between input bitstreams highly depends on the design of pseudo- RNGs and the initial seeds used in different RNG instances.
? We used the curve-fitting mode of our annealing algorithm to synthesize the stochastic versions of these functions.
? It has been demonstrated that the state-mapping network of a digital map in a small-precision digital domain can work as an efficient tool for classifying its structure and coarsely verify its randomness.
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