Complex synchronization structure of an overdamped ratchet with discontinuous periodic forcing.

A deterministic overdamped ratchet driven by a periodic square driving force is shown to display chaotic behavior. The system has neither temporal nor quenched noise but the strong nonlinearity of the driving force produces a very rich bifurcation pattern with synchronized as well as chaotic regions. This pattern disappears if a sinusoidal force replaces the square force. This unexpected behavior is explained by decomposing the system into two exactly solvable subsystems, each with its own characteristic transit time, so that the ratio between the period of the driving force and the transit times can be analyzed. The transition from synchronized to chaotic motion can be explained by means of a one-dimensional Poincaré map. Our results can be experimentally confirmed in a number of systems, including the three-junction superconducting quantum interference devices ratchet, the rocking ratchet effect for cold atoms, and the Josephson vortex ratchet.

Trapping mechanism in overdamped ratchets with quenched noise.

A trapping mechanism is observed and proposed as the origin of the anomalous behavior recently discovered in transport properties of overdamped ratchets subject to an external oscillatory drive in the presence of quenched noise. In particular, this mechanism is shown to appear whenever the quenched disorder strength is greater than a threshold value. The minimum disorder strength required for the existence of traps is determined by studying the trap structure in a disorder configuration space. An approximation to the trapping probability density function in a disordered region of finite length included in an otherwise perfect ratchet lattice is obtained. The mean velocity of the particles and the diffusion coefficient are found to have a nonmonotonic dependence on the quenched noise strength due to the presence of the traps.

Extracting features of Gaussian self-similar stochastic processes via the Bandt-Pompe approach.

By recourse to appropriate information theory quantifiers (normalized Shannon entropy and Martín-Plastino-Rosso intensive statistical complexity measure), we revisit the characterization of Gaussian self-similar stochastic processes from a Bandt-Pompe viewpoint. We show that the ensuing approach exhibits considerable advantages with respect to other treatments. In particular, clear quantifiers gaps are found in the transition between the continuous processes and their associated noises.

Chaotic dynamics and control of deterministic ratchets.

Deterministic ratchets, in the inertial and also in the overdamped limit, have a very complex dynamics, including chaotic motion. This deterministically induced chaos mimics, to some extent, the role of noise, changing, on the other hand, some of the basic properties of thermal ratchets; for example, inertial ratchets can exhibit multiple reversals in the current direction. The direction depends on the amount of friction and inertia, which makes it especially interesting for technological applications such as biological particle separation. We overview in this work different strategies to control the current of inertial ratchets. The control parameters analysed are the strength and frequency of the periodic external force, the strength of the quenched noise that models a non-perfectly-periodic potential, and the mass of the particles. Control mechanisms are associated with the fractal nature of the basins of attraction of the mean velocity attractors. The control of the overdamped motion of noninteracting particles in a rocking periodic asymmetric potential is also reviewed. The analysis is focused on synchronization of the motion of the particles with the external sinusoidal driving force. Two cases are considered: a perfect lattice without disorder and a lattice with noncorrelated quenched noise. The amplitude of the driving force and the strength of the quenched noise are used as control parameters.

Chaos in kicked ratchets.

We present a minimal one-dimensional deterministic continuous dynamical system that exhibits chaotic behavior and complex transport properties. Our model is an overdamped rocking ratchet with finite dissipation, that is periodically kicked with a δ function driving force, without finite inertia terms or temporal or spatial stochastic forces. To our knowledge this is the simplest model reported in the literature for a ratchet, with this complex behavior. We develop an analytical approach that predicts many key features of the system, such as current reversals, as well as the presence of chaotic behavior and bifurcation. Our analytical approach allows us to study the transition from regular to chaotic motion as well as a tangent bifurcation associated with this transition. We show that our approach can be easily extended to other types of periodic driving forces. The square wave is shown as an example.

Quantifiers for randomness of chaotic pseudo-random number generators.

We deal with randomness quantifiers and concentrate on their ability to discern the hallmark of chaos in time series used in connection with pseudo-random number generators (PRNGs). Workers in the field are motivated to use chaotic maps for generating PRNGs because of the simplicity of their implementation. Although there exist very efficient general-purpose benchmarks for testing PRNGs, we feel that the analysis provided here sheds additional didactic light on the importance of the main statistical characteristics of a chaotic map, namely (i) its invariant measure and (ii) the mixing constant. This is of help in answering two questions that arise in applications: (i) which is the best PRNG among the available ones? and (ii) if a given PRNG turns out not to be good enough and a randomization procedure must still be applied to it, which is the best applicable randomization procedure? Our answer provides a comparative analysis of several quantifiers advanced in the extant literature.

Distinguishing noise from chaos.

Chaotic systems share with stochastic processes several properties that make them almost undistinguishable. In this communication we introduce a representation space, to be called the complexity-entropy causality plane. Its horizontal and vertical axis are suitable functionals of the pertinent probability distribution, namely, the entropy of the system and an appropriate statistical complexity measure, respectively. These two functionals are evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the system. Several well-known model-generated time series, usually regarded as being of either stochastic or chaotic nature, are analyzed so as to illustrate the approach. The main achievement of this communication is the possibility of clearly distinguishing between them in our representation space, something that is rather difficult otherwise.