Hugo L. D. de S. Cavalcante

Predictability and suppression of extreme events in a chaotic system.

In many complex systems, large events are believed to follow power-law, scale-free probability distributions so that the extreme, catastrophic events are unpredictable. Here, we study coupled chaotic oscillators that display extreme events. The mechanism responsible for the rare, largest events makes them distinct, and their distribution deviates from a power law. On the basis of this mechanism identification, we show that it is possible to forecast in real time an impending extreme event. Once forecasted, we also show that extreme events can be suppressed by applying tiny perturbations to the system.

On the origin of chaos in autonomous Boolean networks

We undertake a systematic study of the dynamics of Boolean networks to determine the origin of chaos observed in recent experiments. Networks with nodes consisting of ideal logic gates are known to display either steady states, periodic behaviour or an ultraviolet catastrophe where the number of logic-transition events circulating in the network per unit time grows as a power law. In an experiment, the non-ideal behaviour of the logic gates prevents the ultraviolet catastrophe and may lead to deterministic chaos. We identify certain non-ideal features of real logic gates that enable chaos in experimental networks. We find that short-pulse rejection and asymmetry between the logic states tend to engender periodic behaviour, at least for the simplest networks. On the other hand, we find that a memory effect termed ‘degradation’ can generate chaos. Our results strongly suggest that deterministic chaos can be expected in a large class of experimental Boolean-like networks. Such devices may find application in a variety of technologies requiring fast complex waveforms or flat power spectra, and can be used as a test-bed for fundamental studies of real-world Boolean-like networks.

Power law periodicity in the tangent bifurcations of the logistic map

Numerical studies were carried out for the average of the logistic map on the tangent bifurcations from chaos into periodic windows. A critical exponent of 12 is found on the average amplitude as one approaches the transition. Additionally, the averages oscillate with a period that decreases with the same exponent. This Power Law Periodicity is related to the reinjection mechanism of the map. The undulations appear at control parameter values much earlier than the values where the critical exponent of the bifurcation shows significant changes in the average amplitude.

Subwavelength Position Sensing Using Nonlinear Feedback and Wave Chaos

We demonstrate a position-sensing technique that relies on the inherent sensitivity of chaos, where we illuminate a subwavelength object with a complex structured radio-frequency field generated using wave chaos and nonlinear feedback. We operate the system in a quasiperiodic state and analyze changes in the frequency content of the scalar voltage signal in the feedback loop. This allows us to extract the objects position with a one-dimensional resolution of ~λ/10,000 and a two-dimensional resolution of ~λ/300, where λ is the shortest wavelength of the illuminating source.

Experimental bifurcations and homoclinic chaos in a laser with a saturable absorber

The shape and the peak values of the pulses from a passive Q-switching CO2 laser with SF6 as saturable absorber were detected while the laser was tuned in frequency across a longitudinal mode. A succession of stability windows, typical for bifurcation diagrams in the homoclinic scenario, was observed and the widths of those windows were measured. The expansion rate of the undulations in individual pulses was also obtained and compared to Floquets multipliers given by the ratio of widths in consecutive windows. The dynamics is consistent with a homoclinic tangency to a periodic orbit.

Bifurcations and averages in the logistic map

Bifurcations in the logistic map are studied numerically by inspection of the average of a long iterated series. The tangent bifurcations, at the entrance of the periodic windows, and the crisis, when chaotic bands merge, show distinct power law scalings for these statistical properties.

Quantum scaling laws in the onset of dynamical delocalization

By submitting a cloud of cold caesium atoms to a periodically pulsed standing wave, we experimentally realized a quantum system presenting a dynamic that is chaotic in the classical limit called the Kicked Rotor. Such a system presents a phenomenon called dynamical localization (DL). DL is the suppression of the classical chaotic energy growth by quantum interferences due to long range coherence in momentum space. After a breaktime, the quantum momentum distribution is frozen to a steady state and the energy is stuck to an asymptotic value.

Bifurcations and averages in the homoclinic chaos of a laser with a saturable absorber

The dynamical bifurcations of a laser with a saturable absorber were calculated, with the 3–2 level model, as a function of the gain parameter. The average power of the laser is shown to have specific behavior at bifurcations. The succession of periodic–chaotic windows, known to occur in the homoclinic chaos, was studied numerically. A critical exponent of 12 is found at the tangent bifurcations from chaotic into periodic pulsations.

Tunable power law in the desynchronization events of coupled chaotic electronic circuits

We study the statistics of the amplitude of the synchronization error in chaotic electronic circuits coupled through linear feedback. Depending on the coupling strength, our system exhibits three qualitatively different regimes of synchronization: weak coupling yields independent oscillations; moderate to strong coupling produces a regime of intermittent synchronization known as attractor bubbling; and stronger coupling produces complete synchronization. In the regime of moderate coupling, the probability distribution for the sizes of desynchronization events follows a power law, with an exponent that can be adjusted by changing the coupling strength. Such power-law distributions are interesting, as they appear in many complex systems. However, most of the systems with such a behavior have a fixed value for the exponent of the power law, while here we present an example of a system where the exponent of the power law is easily tuned in real time.

Measurement of the Kerr nonlinear refractive index of Cs vapor

Atomic vapors are systems well suited for studies of opticalnonlinearities. First of all, they are easy to saturate, whichenables the observation of nonlinear effects with low intensitycontinuous-wave laser light [1,2]. At the same time, atomicvapors are damage-free which is important, for instance, forfilamentation studies [3]. Second, as the resonances are sharpthe nonlinear parameters can be easily modified by finelytuningthefrequencynearoracrossaresonance[4].Thisallowsto play with the relative contributions of linear and nonlineareffects by changing the laser wavelength. Third, atomicsystems allow for a variety of level schemes exploring fine,hyperfine, and Zeeman levels such as two-level systems [5,6], three-level schemes [4], double- four-level schemes [7,8],five-level schemes [9], and so on. Fourth, in most experiments,when one can ignore radiation trapping and collisional effects,atomic vapors behave as locally saturable media and are thuseasy to model [10].As atomic vapors are isotropic media, the first nonlinearcontribution to the polarization is a third-order term in theelectric field (