Antônio Endler

Noise-enhanced trapping in chaotic scattering

We show that noise enhances the trapping of trajectories in scattering systems. In fully chaotic systems, the decay rate can decrease with increasing noise due to a generic mismatch between the noiseless escape rate and the value predicted by the Liouville measure of the exit set. In Hamiltonian systems with mixed phase space we show that noise leads to a slower algebraic decay due to trajectories performing a random walk inside Kolmogorov-Arnold-Moser islands. We argue that these noise-enhanced trapping mechanisms exist in most scattering systems and are likely to be dominant for small noise intensities, which is confirmed through a detailed investigation in the Hénon map. Our results can be tested in fluid experiments, affect the fractal Weyls law of quantum systems, and modify the estimations of chemical reaction rates based on phase-space transition state theory.

Nonlinear model for thermal effects in free-electron lasers

In the present work, we extend results of a previous paper [Peter et al., Phys. Plasmas 20, 12 3104 (2013)] and develop a semi-analytical model to account for thermal effects on the nonlinear dynamics of the electron beam in free-electron lasers. We relax the condition of a cold electron beam but still use the concept of compressibility, now associated with a warm beam model, to evaluate the time scale for saturation and the peak laser intensity in high-gain regimes. Although vanishing compressibilites and the associated divergent densities are absent in warm models, a series of discontinuities in the electron density precede the saturation process. We show that full wave-particle simulations agree well with the predictions of the model.

Stochastic perturbations in open chaotic systems : Random versus noisy maps

We investigate the effects of random perturbations on fully chaotic open systems. Perturbations can be applied to each trajectory independently (white noise) or simultaneously to all trajectories (random map). We compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time-dependent conditionally-map-invariant measure. For the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map. In random maps we show that the escape rate κ and dimensions D of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation. We discuss the accuracy (bias) and precision (variance) of finite-size estimators of κ and D, and show that the improvement of the precision of the estimations with the number of trajectories N is extremely slow ([proportionality]1/lnN). We also argue that the finite-size D estimators are typically biased. General theoretical results are combined with analytical calculations and numerical simulations in area-preserving baker maps.

Relaxation of intense inhomogeneous charged beams

This work analyzes the dynamics of inhomogeneous, magnetically focused high intensity beams of charged particles. Initial inhomogeneities lead to density waves propagating transversely in the beam core, and the presence of transverse waves eventually results in particle scattering. Particle scattering off waves in the beam core ultimately generates a halo of particles with concomitant emittance growth. Emittance growth indicates a beam relaxing to its final stationary state, and the purpose of the present paper is to describe halo and emittance in terms of test particles moving under the action of the inhomogeneous beam. To this end an average Lagrangian approach for the beam is developed. This approach, aided by the use of conserved quantities, produces results in nice agreement with those obtained with full N-particle numerical simulations.

Mixing and space-charge effects in free-electron lasers

The present work revisits the subjects of mixing, saturation, and space-charge effects in free-electron lasers. Use is made of the compressibility factor, which proves to be a helpful tool in the related systems of charged beams confined by static magnetic fields. The compressibility allows to perform analytical estimates of the elapsed time until the onset of mixing, which in turn allows to estimate the saturated amplitude of the radiation field. In addition, the compressibility helps to pinpoint space-charge effects and the corresponding transition from Compton to Raman regimes.

Extreme and superextreme events in a loss-modulated CO2 laser : nonlinear resonance route and precursors

We investigate the occurrence of extreme and rare events, i.e., giant and rare light pulses, in a periodically modulated CO_{2} laser model. Due to nonlinear resonant processes, we show a scenario of interaction between chaotic bands of different orders, which may lead to the formation of extreme and rare events. We identify a crisis line in the modulation parameter space, and we show that, when the modulation amplitude increases, remaining in the vicinity of the crisis, some statistical properties of the laser pulses, such as the average and dispersion of amplitudes, do not change much, whereas the amplitude of extreme events grows enormously, giving rise to extreme events with much larger deviations than usually reported, with a significant probability of occurrence, i.e., with a long-tailed non-Gaussian distribution. We identify recurrent regular patterns, i.e., precursors, that anticipate the emergence of extreme and rare events, and we associate these regular patterns with unstable periodic orbits embedded in a chaotic attractor. We show that the precursors may or may not lead to the emergence of extreme events. Thus, we compute the probability of success or failure (false alarm) in the prediction of the extreme events, once a precursor is identified in the deterministic time series. We show that this probability depends on the accuracy with which the precursor is identified in the laser intensity time series.

Period four stability and multistability domains for the Hénon map

We report the exact analytical expression of the surface W4(a,b;λ)=0 defining stability domains for period-4 motions in the Henon map, valid for arbitrary eigenvalues λ and parameters a and b. For λ=+1 (fold bifurcations) the expression reproduces all previous results and gives a new one. For λ=−1 (flip bifurcations) it gives analytically the missing boundary needed for the rigorous delimitation of all period-4 stability domains and for the investigation of the arithmetic nature of parameters and trajectories.

From Mandelbrot-like sets to Arnold tongues

A transition from Mandelbrot-like sets to Arnold tongues is characterized via a coupling of two non-identical quadratic maps proposed by us. A two-dimensional parameter-space considering the parameters of the individual quadratic maps was used to demonstrate numerically the event. The location of the parameter sets where Naimark-Sacker bifurcations occur, which is exactly the place where Arnold tongues of arbitrary periods are born, was computed analytically.