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Subject of this work is the investigation of universal scaling laws which are observed in coupled chaotic systems. Progress is made by replacing the chaotic fluctuations in the perturbation dynamics by stochastic processes. First, a continuous-time stochastic model for weakly coupled chaotic systems is introduced to study the scaling of the Lyapunov exponents with the coupling strength (coupling sensitivity of chaos). By means of the the Fokker-Planck equation scaling relations are derived, which are confirmed by results of numerical simulations. Next, the new effect of avoided crossing of Lyapunov exponents of weakly coupled disordered chaotic systems is described, which is qualitatively similar to the energy level repulsion in quantum systems. Using the scaling relations obtained for the coupling sensitivity of chaos, an asymptotic expression for the distribution function of small spacings between Lyapunov exponents is derived and compared with results of numerical simulations. Finally, the synchronization transition in strongly coupled spatially extended chaotic systems is shown to resemble a continuous phase transition, with the coupling strength and the synchronization error as control and order parameter, respectively. Using results of numerical simulations and theoretical considerations in terms of a multiplicative noise partial differential equation, the universality classes of the observed two types of transition are determined (Kardar-Parisi-Zhang equation with saturating term, directed percolation).
We study two coupled spatially extended dynamical systems which exhibit space-time chaos. The transition to the synchronized state is treated as a nonequilibrium phase transition, where the average synchronization error is the order parameter. The transition in one-dimensional systems is found to be generically in the universality class of the Kardar- Parisi-Zhang equation with a growth-limiting term ("bounded KPZ"). For systems with very strong nonlinearities in the local dynamics, however, the transition is found to be in the universality class of directed percolation.
The behavior of the Lyapunov exponents (LEs) of a disordered system consisting of mutually coupled chaotic maps with different parameters is studied. The LEs are demonstrated to exhibit avoided crossing and level repulsion, qualitatively similar to the behavior of energy levels in quantum chaos. Recent results for the coupling dependence of the LEs of two coupled chaotic systems are used to explain the phenomenon and to derive an approximate expression for the distribution functions of LE spacings. The depletion of the level spacing distribution is shown to be exponentially strong at small values. The results are interpreted in terms of the random matrix theory.
Aims. We present an extensive study of the BL Lac object Mrk 501 based on a data set collected during the multi-instrument campaign spanning from 2009 March 15 to 2009 August 1, which includes, among other instruments, MAGIC, VERITAS, Whipple 10 m, and Fermi-LAT to cover the gamma-ray range from 0.1 GeV to 20 TeV; RXTE and Swift to cover wavelengths from UV to hard X-rays; and GASP-WEBT, which provides coverage of radio and optical wavelengths. Optical polarization measurements were provided for a fraction of the campaign by the Steward and St. Petersburg observatories. We evaluate the variability of the source and interband correlations, the gamma-ray flaring activity occurring in May 2009, and interpret the results within two synchrotron self-Compton (SSC) scenarios. Methods. The multiband variability observed during the full campaign is addressed in terms of the fractional variability, and the possible correlations are studied by calculating the discrete correlation function for each pair of energy bands where the significance was evaluated with dedicated Monte Carlo simulations. The space of SSC model parameters is probed following a dedicated grid-scan strategy, allowing for a wide range of models to be tested and offering a study of the degeneracy of model-to-data agreement in the individual model parameters, hence providing a less biased interpretation than the "single-curve SSC model adjustment" typically reported in the literature. Results. We find an increase in the fractional variability with energy, while no significant interband correlations of flux changes are found on the basis of the acquired data set. The SSC model grid-scan shows that the flaring activity around May 22 cannot be modeled adequately with a one-zone SSC scenario (using an electron energy distribution with two breaks), while it can be suitably described within a two (independent) zone SSC scenario. Here, one zone is responsible for the quiescent emission from the averaged 4.5-month observing period, while the other one, which is spatially separated from the first, dominates the flaring emission occurring at X-rays and very-high-energy (> 100 GeV, VHE) gamma-rays. The flaring activity from May 1, which coincides with a rotation of the electric vector polarization angle (EVPA), cannot be satisfactorily reproduced by either a one-zone or a two-independent-zone SSC model, yet this is partially affected by the lack of strictly simultaneous observations and the presence of large flux changes on sub-hour timescales (detected at VHE gamma rays). Conclusions. The higher variability in the VHE emission and lack of correlation with the X-ray emission indicate that, at least during the 4.5-month observing campaign in 2009, the highest energy (and most variable) electrons that are responsible for the VHE gamma rays do not make a dominant contribution to the similar to 1 keV emission. Alternatively, there could be a very variable component contributing to the VHE gamma-ray emission in addition to that coming from the SSC scenario. The studies with our dedicated SSC grid-scan show that there is some degeneracy in both the one-zone and the two-zone SSC scenarios probed, with several combinations of model parameters yielding a similar model-to-data agreement, and some parameters better constrained than others. The observed gamma-ray flaring activity, with the EVPA rotation coincident with the first gamma-ray flare, resembles those reported previously for low frequency peaked blazars, hence suggesting that there are many similarities in the flaring mechanisms of blazars with different jet properties.
Aims. We aim to characterize the multiwavelength emission from Markarian 501 (Mrk 501), quantify the energy-dependent variability, study the potential multiband correlations, and describe the temporal evolution of the broadband emission within leptonic theoretical scenarios. Methods. We organized a multiwavelength campaign to take place between March and July of 2012. Excellent temporal coverage was obtained with more than 25 instruments, including the MAGIC, FACT and VERITAS Cherenkov telescopes, the instruments on board the Swift and Fermi spacecraft, and the telescopes operated by the GASP-WEBT collaboration. Results. Mrk 501 showed a very high energy (VHE) gamma-ray flux above 0.2 TeV of similar to 0.5 times the Crab Nebula flux (CU) for most of the campaign. The highest activity occurred on 2012 June 9, when the VHE flux was similar to 3 CU, and the peak of the high-energy spectral component was found to be at similar to 2 TeV. Both the X-ray and VHE gamma-ray spectral slopes were measured to be extremely hard, with spectral indices <2 during most of the observing campaign, regardless of the X-ray and VHE flux. This study reports the hardest Mrk 501 VHE spectra measured to date. The fractional variability was found to increase with energy, with the highest variability occurring at VHE. Using the complete data set, we found correlation between the X-ray and VHE bands; however, if the June 9 flare is excluded, the correlation disappears (significance <3 sigma) despite the existence of substantial variability in the X-ray and VHE bands throughout the campaign. Conclusions. The unprecedentedly hard X-ray and VHE spectra measured imply that their low- and high-energy components peaked above 5 keV and 0.5 TeV, respectively, during a large fraction of the observing campaign, and hence that Mrk 501 behaved like an extreme high-frequency-peaked blazar (EHBL) throughout the 2012 observing season. This suggests that being an EHBL may not be a permanent characteristic of a blazar, but rather a state which may change over time. The data set acquired shows that the broadband spectral energy distribution (SED) of Mrk 501, and its transient evolution, is very complex, requiring, within the framework of synchrotron self-Compton (SSC) models, various emission regions for a satisfactory description. Nevertheless the one-zone SSC scenario can successfully describe the segments of the SED where most energy is emitted, with a significant correlation between the electron energy density and the VHE gamma-ray activity, suggesting that most of the variability may be explained by the injection of high-energy electrons. The one-zone SSC scenario used reproduces the behavior seen between the measured X-ray and VHE gamma-ray fluxes, and predicts that the correlation becomes stronger with increasing energy of the X-rays.
Context. The large jet kinetic power and non-thermal processes occurring in the microquasar SS 433 make this source a good candidate for a very high-energy (VHE) gamma-ray emitter. Gamma-ray fluxes above the sensitivity limits of current Cherenkov telescopes have been predicted for both the central X-ray binary system and the interaction regions of SS 433 jets with the surrounding W50 nebula. Non-thermal emission at lower energies has been previously reported, indicating that efficient particle acceleration is taking place in the system. Aims. We explore the capability of SS 433 to emit VHE gamma rays during periods in which the expected flux attenuation due to periodic eclipses (P-orb similar to 13.1 days) and precession of the circumstellar disk (P-pre similar to 162 days) periodically covering the central binary system is expected to be at its minimum. The eastern and western SS 433/W50 interaction regions are also examined using the whole data set available. We aim to constrain some theoretical models previously developed for this system with our observations. Methods. We made use of dedicated observations from the Major Atmospheric Gamma Imaging Cherenkov telescopes (MAGIC) and High Energy Spectroscopic System (H.E.S.S.) of SS 433 taken from 2006 to 2011. These observation were combined for the first time and accounted for a total effective observation time of 16.5 h, which were scheduled considering the expected phases of minimum absorption of the putative VHE emission. Gamma-ray attenuation does not affect the jet/medium interaction regions. In this case, the analysis of a larger data set amounting to similar to 40-80 h, depending on the region, was employed. Results. No evidence of VHE gamma-ray emission either from the central binary system or from the eastern/western interaction regions was found. Upper limits were computed for the combined data set. Differential fluxes from the central system are found to be less than or similar to 10(-12)-10(-13) TeV-1 cm(-2) s(-1) in an energy interval ranging from similar to few x 100 GeV to similar to few TeV. Integral flux limits down to similar to 10(-12)-10(-13) ph cm(-2) s(-1) and similar to 10(-13)-10(-14) ph cm(-2) s(-1) are obtained at 300 and 800 GeV, respectively. Our results are used to place constraints on the particle acceleration fraction at the inner jet regions and on the physics of the jet/medium interactions. Conclusions. Our findings suggest that the fraction of the jet kinetic power that is transferred to relativistic protons must be relatively small in SS 433, q(p) <= 2.5 x 10(-5), to explain the lack of TeV and neutrino emission from the central system. At the SS 433/W50 interface, the presence of magnetic fields greater than or similar to 10 mu G is derived assuming a synchrotron origin for the observed X-ray emission. This also implies the presence of high-energy electrons with E-e up to 50 TeV, preventing an efficient production of gamma-ray fluxes in these interaction regions.
In the present work, we study wave phenomena in strongly nonlinear lattices. Such lattices are characterized by the absence of classical linear waves. We demonstrate that compactons – strongly localized solitary waves with tails decaying faster than exponential – exist and that they play a major role in the dynamics of the system under consideration. We investigate compactons in different physical setups. One part deals with lattices of dispersively coupled limit cycle oscillators which find various applications in natural sciences such as Josephson junction arrays or coupled Ginzburg-Landau equations. Another part deals with Hamiltonian lattices. Here, a prominent example in which compactons can be found is the granular chain. In the third part, we study systems which are related to the discrete nonlinear Schrödinger equation describing, for example, coupled optical wave-guides or the dynamics of Bose-Einstein condensates in optical lattices. Our investigations are based on a numerical method to solve the traveling wave equation. This results in a quasi-exact solution (up to numerical errors) which is the compacton. Another ansatz which is employed throughout this work is the quasi-continuous approximation where the lattice is described by a continuous medium. Here, compactons are found analytically, but they are defined on a truly compact support. Remarkably, both ways give similar qualitative and quantitative results. Additionally, we study the dynamical properties of compactons by means of numerical simulation of the lattice equations. Especially, we concentrate on their emergence from physically realizable initial conditions as well as on their stability due to collisions. We show that the collisions are not exactly elastic but that a small part of the energy remains at the location of the collision. In finite lattices, this remaining part will then trigger a multiple scattering process resulting in a chaotic state.
We study localized traveling waves and chaotic states in strongly nonlinear one-dimensional Hamiltonian lattices. We show that the solitary waves are superexponentially localized and present an accurate numerical method allowing one to find them for an arbitrary nonlinearity index. Compactons evolve from rather general initially localized perturbations and collide nearly elastically. Nevertheless, on a long time scale for finite lattices an extensive chaotic state is generally observed. Because of the system's scaling, these dynamical properties are valid for any energy.
Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case, the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance; i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed.