Spectrum of the Andreev Billiard and Giant Fluctuations of the Ehrenfest Time

The density of states in the semiclassical Andreev billiard is theoretically studied and shown to be determined by the fluctuations of the classical Lyapunov exponent $\lambda$. The rare trajectories with a small value of $\lambda$ give rise to an anomalous increase of the Ehrenfest time $\tau_E\approx |\ln\hbar|/\lambda$ and, consequently, to the appearance of Andreev levels with small excitation energy. The gap in spectrum is obtained and fluctuations of the value of the gap due to different positions of superconducting lead are considered.Comment: 4 pages, 3 figure

Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution

We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's complete stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.Comment: 28 pages, 7 figures; submitted to Phys. Rev. E Added comment

On the equality of Hausdorff and box counting dimensions

By viewing the covers of a fractal as a statistical mechanical system, the exact capacity of a multifractal is computed. The procedure can be extended to any multifractal described by a scaling function to show why the capacity and Hausdorff dimension are expected to be equal.Comment: CYCLER Paper 93mar001 Latex file with 3 PostScript figures (needs psfig.sty

Efficient time series detection of the strong stochasticity threshold in Fermi-Pasta-Ulam oscillator lattices

In this work we study the possibility of detecting the so-called strong stochasticity threshold, i.e. the transition between weak and strong chaos as the energy density of the system is increased, in anharmonic oscillator chains by means of the 0-1 test for chaos. We compare the result of the aforementioned methodology with the scaling behavior of the largest Lyapunov exponent computed by means of tangent space dynamics, that has so far been the most reliable method available to detect the strong stochasticity threshold. We find that indeed the 0-1 test can perform the detection in the range of energy density values studied. Furthermore, we determined that conventional nonlinear time series analysis methods fail to properly compute the largest Lyapounov exponent even for very large data sets, whereas the computational effort of the 0-1 test remains the same in the whole range of values of the energy density considered with moderate size time series. Therefore, our results show that, for a qualitative probing of phase space, the 0-1 test can be an effective tool if its limitations are properly taken into account.Comment: 5 pages, 2 figures; accepted for publication in Physical Review

Quantum chaos of a mixed, open system of kicked cold atoms

The quantum and classical dynamics of particles kicked by a gaussian attractive potential are studied. Classically, it is an open mixed system (the motion in some parts of the phase space is chaotic, and in some parts it is regular). The fidelity (Lochshmidt echo) is found to exhibit oscillations that can be determined from classical considerations but are sensitive to phase space structures that are smaller than Planck's constant. Families of quasi-energies are determined from classical phase space structures. Substantial differences between the classical and quantum dynamics are found for time dependent scattering. It is argued that the system can be experimentally realized by cold atoms kicked by a gaussian light beam.Comment: 19 pages, 21 figures, (accepted for publication in Phys. Rev. E

Rectenna system design

The function of the rectenna in the solar power satellite system is described and the basic design choices based on the desired microwave field concentration and ground clearance requirements are given. One important area of concern, from the EMI point of view, harmonic reradiation and scattering from the rectenna is also designed. An optimization of a rectenna system design to minimize costs was performed. The rectenna cost breakdown for a 56 w installation is given as an example

Classical diffusion in double-delta-kicked particles

We investigate the classical chaotic diffusion of atoms subjected to {\em pairs} of closely spaced pulses (`kicks) from standing waves of light (the $2\delta$-KP). Recent experimental studies with cold atoms implied an underlying classical diffusion of type very different from the well-known paradigm of Hamiltonian chaos, the Standard Map. The kicks in each pair are separated by a small time interval $\epsilon \ll 1$, which together with the kick strength $K$, characterizes the transport. Phase space for the $2\delta$-KP is partitioned into momentum `cells' partially separated by momentum-trapping regions where diffusion is slow. We present here an analytical derivation of the classical diffusion for a $2\delta$-KP including all important correlations which were used to analyze the experimental data. We find a new asymptotic ($t \to \infty$) regime of `hindered' diffusion: while for the Standard Map the diffusion rate, for $K \gg 1$, $D \sim K^2/2[1- J_2(K)..]$ oscillates about the uncorrelated, rate $D_0 =K^2/2$, we find analytically, that the $2\delta$-KP can equal, but never diffuses faster than, a random walk rate. We argue this is due to the destruction of the important classical `accelerator modes' of the Standard Map. We analyze the experimental regime $0.1\lesssim K\epsilon \lesssim 1$, where quantum localisation lengths $L \sim \hbar^{-0.75}$ are affected by fractal cell boundaries. We find an approximate asymptotic diffusion rate $D\propto K^3\epsilon$, in correspondence to a $D\propto K^3$ regime in the Standard Map associated with 'golden-ratio' cantori.Comment: 14 pages, 10 figures, error in equation in appendix correcte

Quantifying Spatiotemporal Chaos in Rayleigh-B\'enard Convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-B\'enard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading order Lyapunov exponent and we quantify the details of their response to the dynamics of defects. The leading order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary dominated dynamics to bulk dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics

Ordering of small particles in one-dimensional coherent structures by time-periodic flows

Small particles transported by a fluid medium do not necessarily have to follow the flow. We show that for a wide class of time-periodic incompressible flows inertial particles have a tendency to spontaneously align in one-dimensional dynamic coherent structures. This effect may take place for particles so small that often they would be expected to behave as passive tracers and be used in PIV measurement technique. We link the particle tendency to form one-dimensional structures to the nonlinear phenomenon of phase locking. We propose that this general mechanism is, in particular, responsible for the enigmatic formation of the `particle accumulation structures' discovered experimentally in thermocapillary flows more than a decade ago and unexplained until now

The effect of short ray trajectories on the scattering statistics of wave chaotic systems

In many situations, the statistical properties of wave systems with chaotic classical limits are well-described by random matrix theory. However, applications of random matrix theory to scattering problems require introduction of system specific information into the statistical model, such as the introduction of the average scattering matrix in the Poisson kernel. Here it is shown that the average impedance matrix, which also characterizes the system-specific properties, can be expressed in terms of classical trajectories that travel between ports and thus can be calculated semiclassically. Theoretical results are compared with numerical solutions for a model wave-chaotic system