José Carlos Sartorelli

Chaotic behavior in bubble formation dynamics

We constructed an experimental apparatus to study the dynamics of the formation of air bubbles in a submerged nozzle in a water/glycerin solution inside a cylindrical tube. The delay time between successive bubbles was measured with a laser-photodiode system. It was observed bifurcations, chaotic behavior, and sudden changes in a periodic regime as a function of the decreasing air pressure in a reservoir. We also observed dynamical effects by applying a sound wave tuned to the fundamental frequency of the air column above the solution. As a function of the sound wave amplitude, we obtained a limit cycle, a flip bifurcation, chaotic behavior, and the synchronization of the bubbling with sound wave frequency. We related some of the different dynamical behaviors to coalescent effects and bubble sizes.

Long-range anticorrelations and non-Gaussian behavior of a leaky faucet

We find that intervals between successive drops from a leaky faucet display scale-invariant, long-range anticorrelations characterized by the same exponents of heart beat-to-beat intervals of healthy subjects. This behavior is also confirmed by numerical simulations on lattice and it is faucet-width- and flow-rate-independent. The histogram for the drop intervals is also well described by a Levy distribution with the same index for both histograms of healthy and diseased subjects. This additional result corroborates the evidence for similarities between leaky faucets and healthy hearts underlying dynamics.

The circle map dynamics in air bubble formation

Abstract We studied the air bubbles formation in a submerged nozzle in a water/glycerol solution inside a cylindrical tube, submitted to a sound wave perturbation, whose amplitude is a parameter of control. It was experimentally observed quasiperiodicity, transition from quasiperiodicity to chaos, routes to chaos via period doubling cascade according to the values of Ω=f s /f b , where fs is the sound wave frequency and fb is the bubbling rate. Our data can be explained by a two-dimensional circle map dynamics. We simulated some bifurcation diagrams as well as some reconstructed attractors with amazing results.

Mutual Information Rate and Bounds for It

The amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems. This quantity, known as the mutual information rate (MIR), is calculated from the mutual information, which is rigorously defined only for random systems. Moreover, the definition of mutual information is based on probabilities of significant events. This work offers a simple alternative way to calculate the MIR in dynamical (deterministic) networks or between two time series (not fully deterministic), and to calculate its upper and lower bounds without having to calculate probabilities, but rather in terms of well known and well defined quantities in dynamical systems. As possible applications of our bounds, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators.

Inferring statistical complexity in the dripping faucet experiment

We have used topological e-machine reconstruction to calculate statistical complexity associated to period five orbits in the dripping faucet experiment. Basic structures related to the periodic movement and to peripheral tracks in the corresponding directed graphs were identified.

Hénon-like attractor in air bubble formation

Abstract We studied the formation of air bubbles in a submerged nozzle in a water/glycerol solution inside a cylindrical tube, submitted to a sound wave perturbation. It was observed a route to chaos via period doubling as a function of the sound wave amplitude. We applied metrical as well as topological characterization to some chaotic attractors. We localized a flip saddle, and we also could establish relations to a Henon-like dynamics with the construction of symbolic planes.

Simulations in a dripping faucet experiment

Abstract The profiles of two experimental attractors were simulated by using a simple one-dimensional spring-mass model. Some peculiar behaviors observed in experimental bifurcation diagrams (in short ranges of dripping rate variation) were emulated by combining two quadratic maps (a kind of coupling) in two different ways: parallel combination with non-interacting maps; and series combination with strongly interacting maps. The choice of each kind of combination was suggested by the own characteristics of each experimental bifurcation diagram.

Non-transitive maps in phase synchronization

Abstract Concepts from Ergodic Theory are used to describe the existence of special non-transitive maps in attractors of phase synchronous chaotic oscillators. In particular, it is shown that, for a class of phase-coherent oscillators, these special maps imply phase synchronization. We illustrate these ideas in the sinusoidally forced Chua’s circuit and two coupled Rossler oscillators. Furthermore, these results are extended to other coupled chaotic systems. In addition, a phase for a chaotic attractor is defined from the tangent vector of the flow. Finally, it is discussed how these maps can be used for the real-time detection of phase synchronization in experimental systems.

A SCALE LAW IN A DRIPPING FAUCET

Abstract The evolution to a period-1 motion after an inverse secondary Hopf bifurcation, at f0 = 39.976 drops/s, was characterized by the autocorrelation function. The amplitude of the autocorrelation function for f > f0 decays exponentially; its characteristic correlation drop scales with | (f − f c ) f c | γ , where fc = 39.897 drops/sγ = −2.28 ± 0.03.

Period-adding bifurcations and chaos in a bubble column

We obtained period-adding bifurcations in a bubble formation experiment. Using the air flow rate as the control parameter in this experiment, the bubble emission from the nozzle in a viscous fluid undergoes from single bubbling to a sequence of periodic bifurcations of k to k+1 periods, occasionally interspersed with some chaotic regions. Our main assumption is that this period-adding bifurcation in bubble formation depends on flow rate variations in the chamber under the nozzle. This assumption was experimentally tested by placing a tube between the air reservoir and the chamber under the nozzle in the bubble column experiment. By increasing the tube length, more period-adding bifurcations were observed. We associated two main types of bubble growth to the flow rate fluctuations inside the chamber for different bubbling regimes. We also studied the properties of piecewise nonlinear maps obtained from the experimental reconstructed attractors, and we concluded that this experiment is a spatially extended system.