Chaotic System

In the paper we propose a new method of constructing cryptosystems utilising a nonpredictability property of discrete chaotic systems. We formulate the requirements for such systems to assure their safety. We also give examples of practical realisation of chaotic cryptosystems, using a generalisation of the method presented in [7]. The proposed algorithm of encryption and decryption is based on multiple iteration of a certain dynamical chaotic system. We assume that some part of the initial condition is a plain message. As the secret key we assume the system parameter(s) and additionally another part of the initial condition.

Emergent behaviors are in the focus of recent research interest. It is then of considerable importance to investigate what optimizations suit the learning and prediction of chaotic systems, the putative candidates for emergence. We have compared L1 and L2 regularizations on predicting chaotic time series using linear recurrent neural networks. The internal representation and the weights of the networks were optimized in a unifying framework. Computational tests on different problems indicate considerable advantages for the L1 regularization: It had considerably better learning time and better interpolating capabilities. We shall argue that optimization viewed as a maximum likelihood estimation justifies our results, because L1 regularization fits heavy-tailed distributions -- an apparently general feature of emergent systems -- better.

A simplified version of the stochastic Fermi-Ulam model is investigated in order to elucidate the effect of a class of rare low-velocity events on the velocity diffusion process and consequently Fermi acceleration. The relative fraction of these events, for sufficiently large times, decreases monotonically with increasing variance of the magnitude of the particle velocity. However, a treatment of the diffusion problem which totally neglects these events, gives rise to a glaring inconsistency associated with the mean value of the magnitude of the velocity in the ensemble. We propose a general scheme for treating the diffusion process in velocity space, which succeeds in capturing the effect of the low-velocity events on the diffusion, providing a consistent description of the acceleration process. The present study exemplifies the influence of low-probability events on the transport properties of time-dependent billiards.

Bio-inspired solutions are often applied to solve optimization problems. In this paper the introduction of chaotic systems in Ant Colony Optimization (ACO) algorithms is investigated. The ACO strategy is inspired by the cooperative behavior of food retrieval shown by ants that collectively discover the shortest path between the ant colony and food sources. The optimization problem examined in this work is the well-known Travelling Salesman Problem (TSP), a standard test bench for new combinatory optimization algorithms. The simulation results show that the application of deterministic chaotic signals instead of random sequences is a possible strategy to improve the performances of ACO algorithms.