Branching processes and computational collapse of discretized unimodal mappings

In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping.

Synchronization and Basins of Synchronized in Two-Dimensional Piecewise Maps Via Coupling Three Pieces of One-Dimensional Maps

This paper is devoted to the synchronization of a dynamical system defined by two different coupling versions of two identical piecewise linear bimodal maps. We consider both local and global studies, using different tools as natural transversal Lyapunov exponent, Lyapunov functions, eigenvalues and eigenvectors and numerical simulations. We obtain theoretical results for the existence of synchronization on coupling parameter range. We characterize the synchronization manifold as an attractor and measure the synchronization speed. In one coupling version, we give a necessary and sufficient condition for the synchronization. We study the basins of synchronization and show that, depending upon the type of coupling, they can have very different shapes and are not necessarily constituted by the whole phase space; in some cases, they can be riddled.

Scaling law in Saddle-node bifurcations for one-dimensional maps: A complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.

Load Profiling Tool to Support Smart Grid Operation Scenarios

This paper presents the characterization of high voltage (HV) electric power consumers based on a data clustering approach. The typical load profiles (TLP) are obtained selecting the best partition of a power consumption database among a pool of data partitions produced by several clustering algorithms. The choice of the best partition is supported using several cluster validity indices. The proposed data-mining (DM) based methodology, that includes all steps presented in the process of knowledge discovery in databases (KDD), presents an automatic data treatment application in order to preprocess the initial database in an automatic way, allowing time saving and better accuracy during this phase. These methods are intended to be used in a smart grid environment to extract useful knowledge about customers’ consumption behavior. To validate our approach, a case study with a real database of 185 HV consumers was used.

Measurement of exclusive γγ→ℓ+ℓ− production in proton--proton collisions at s√=7 TeV with the ATLAS detector

This Letter reports a measurement of the exclusive γγ→ℓ+ℓ−(ℓ=e,μ) cross-section in proton--proton collisions at a centre-of-mass energy of 7 TeV by the ATLAS experiment at the LHC, based on an integrated luminosity of 4.6 fb−1. For the electron or muon pairs satisfying exclusive selection criteria, a fit to the dilepton acoplanarity distribution is used to extract the fiducial cross-sections. The cross-section in the electron channel is determined to be σexcl.γγ→e+e−=0.428±0.035(stat.)±0.018(syst.) pb for a phase-space region with invariant mass of the electron pairs greater than 24 GeV, in which both electrons have transverse momentum pT>12 GeV and pseudorapidity |η|<2.4. For muon pairs with invariant mass greater than 20 GeV, muon transverse momentum pT>10 GeV and pseudorapidity |η|<2.4, the cross-section is determined to be σexcl.γγ→μ+μ−=0.628±0.032(stat.)±0.021(syst.) pb. When proton absorptive effects due to the finite size of the proton are taken into account in the theory calculation, the measured cross-sections are found to be consistent with the theory prediction.

Asymptotic flocking dynamics for the kinetic Cucker-Smale model

In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.

Simultaneous cell morphometry and refractive index measurement with dual-wavelength digital holographic microscopy and dye-enhanced dispersion of perfusion medium.

Digital holographic microscopy (DHM) allows optical-path-difference (OPD) measurements with nanometric accuracy. OPD induced by transparent cells depends on both the refractive index (RI) of cells and their morphology. This Letter presents a dual-wavelength DHM that allows us to separately measure both the RI and the cellular thickness by exploiting an enhanced dispersion of the perfusion medium achieved by the utilization of an extracellular dye. The two wavelengths are chosen in the vicinity of the absorption peak of the dye, where the absorption is accompanied by a significant variation of the RI as a function of the wavelength.

Krypton laser photocoagulation induces retinal vascular remodeling rather than choroidal neovascularization.

The purpose of this study is to analyze the retina and choroid response following krypton laser photocoagulation. Ninety-two C57BL6/Sev129 and 32 C57BL/6J, 5-6-week-old mice received one single krypton (630 nm) laser lesion: 50 microm, 0.05 s, 400 mW. On the following day, every day thereafter for 1 week and every 2-3 days for the following 3 weeks, serial sections throughout the lesion were systematically collected and studied. Immunohistology using specific markers or antibodies for glial fibrillary acidic protein (GFAP) (astrocytes, glia and Muller's cells), von Willebrand (vW) (vascular endothelial cells), TUNEL (cells undergoing caspase dependent apoptosis), PCNA (proliferating cell nuclear antigen) p36, CD4 and F4/80 (infiltrating inflammatory and T cells), DAPI (cell nuclei) and routine histology were carried out. Laser confocal microscopy was also performed on flat mounts. Temporal and spatial observations of the created photocoagulation lesions demonstrate that, after a few hours, activated glial cells within the retinal path of the laser beam express GFAP. After 48 h, GFAP-positive staining was also detected within the choroid lesion center. "Movement" of this GFAP-positive expression towards the lasered choroid was preceded by a well-demarcated and localized apoptosis of the retina outer nuclear layer cells within the laser beam path. Later, death of retinal outer nuclear cells and layer thinning at this site was followed by evagination of the inner nuclear retinal layer. Funneling of the entire inner nuclear and the thinned outer nuclear layers into the choroid lesion center was accompanied by "dragging" of the retinal capillaries. Thus, from days 10 to 14 after krypton laser photocoagulation onward, well-formed blood capillaries (of retinal origin) were observed within the lesion. Only a few of the vW-positive capillary endothelial cells stained also for PCNA p36. In the choroid, dilatation of the vascular bed occurred at the vicinity of the photocoagulation site and around it. Confocal microscopy demonstrates that the vessels throughout the path lesion are located within the neuroretina while in the choroid (after separation of the neural retina) only GFAP-positive but no lectin-positive cells can be seen. The involvement of infiltrating inflammatory cells in these remodeling and healing processes remained minimal throughout the study period. During the 4 weeks following krypton laser photocoagulation in the mouse eye, processes of wound healing and remodeling appear to be driven by cells (and vessels) originating from the retina.

Covariant description of kinetic freeze-out through a finite spacelike layer

The problem of freeze-out (FO) in relativistic heavy-ion reactions is addressed. We develop and analyze an idealized one-dimensional model of FO in a finite layer, based on the covariant FO probability. The resulting post FO phase-space distributions are discussed for different FO probabilities and layer thicknesses.

Gauge transformations in the Lagrangian and Hamilton formalisms of generally covariant theories

We study spacetime diffeomorphisms in the Hamiltonian and Lagrangian formalisms of generally covariant systems. We show that the gauge group for such a system is characterized by having generators which are projectable under the Legendre map. The gauge group is found to be much larger than the original group of spacetime diffeomorphisms, since its generators must depend on the lapse function and shift vector of the spacetime metric in a given coordinate patch. Our results are generalizations of earlier results by Salisbury and Sundermeyer. They arise in a natural way from using the requirement of equivalence between Lagrangian and Hamiltonian formulations of the system, and they are new in that the symmetries are realized on the full set of phase space variables. The generators are displayed explicitly and are applied to the relativistic string and to general relativity.

Dimensional reduction and gauge group reduction in Bianchi-type cosmology

In this paper we examine in detail the implementation, with its associated difficulties, of the Killing conditions and gauge fixing into the variational principle formulation of Bianchi-type cosmologies. We address problems raised in the literature concerning the Lagrangian and the Hamiltonian formulations: We prove their equivalence, make clear the role of the homogeneity preserving diffeomorphisms in the phase space approach, and show that the number of physical degrees of freedom is the same in the Hamiltonian and Lagrangian formulations. Residual gauge transformations play an important role in our approach, and we suggest that Poincaré transformations for special relativistic systems can be understood as residual gauge transformations. In the Appendixes, we give the general computation of the equations of motion and the Lagrangian for any Bianchi-type vacuum metric and for spatially homogeneous Maxwell fields in a nondynamical background (with zero currents). We also illustrate our counting of degrees of freedom in an appendix.

Gauge group and reality conditions in Ashtekar's complex formulation of canonical gravity

We discuss reality conditions and the relation between spacetime diffeomorphisms and gauge transformations in Ashtekars complex formulation of general relativity. We produce a general theoretical framework for the stabilization algorithm for the reality conditions, which is different from Diracs method of stabilization of constraints. We solve the problem of the projectability of the diffeomorphism transformations from configuration-velocity space to phase space, linking them to the reality conditions. We construct the complete set of canonical generators of the gauge group in the phase space which includes all the gauge variables. This result proves that the canonical formalism has all the gauge structure of the Lagrangian theory, including the time diffeomorphisms.

Issue of time in generally covariant theories and the Komar-Bergmann approach to observables in general relativity

Diffeomorphism-induced symmetry transformations and time evolution are distinct operations in generally covariant theories formulated in phase space. Time is not frozen. Diffeomorphism invariants are consequently not necessarily constants of the motion. Time-dependent invariants arise through the choice of an intrinsic time, or equivalently through the imposition of time-dependent gauge fixation conditions. One example of such a time-dependent gauge fixing is the Komar-Bergmann use of Weyl curvature scalars in general relativity. An analogous gauge fixing is also imposed for the relativistic free particle and the resulting complete set time-dependent invariants for this exactly solvable model are displayed. In contrast with the free particle case, we show that gauge invariants that are simultaneously constants of motion cannot exist in general relativity. They vary with intrinsic time.

Effects of small surface tension in Hele-Shaw multifinger dynamics: an analytical and numerical study

We study the singular effects of vanishingly small surface tension on the dynamics of finger competition in the Saffman-Taylor problem, using the asymptotic techniques described by Tanveer [Philos. Trans. R. Soc. London, Ser. A 343, 155 (1993)] and Siegel and Tanveer [Phys. Rev. Lett. 76, 419 (1996)], as well as direct numerical computation, following the numerical scheme of Hou, Lowengrub, and Shelley [J. Comput. Phys. 114, 312 (1994)]. We demonstrate the dramatic effects of small surface tension on the late time evolution of two-finger configurations with respect to exact (nonsingular) zero-surface-tension solutions. The effect is present even when the relevant zero-surface-tension solution has asymptotic behavior consistent with selection theory. Such singular effects, therefore, cannot be traced back to steady state selection theory, and imply a drastic global change in the structure of phase-space flow. They can be interpreted in the framework of a recently introduced dynamical solvability scenario according to which surface tension unfolds the structurally unstable flow, restoring the hyperbolicity of multifinger fixed points.

Dynamical Systems approach to Saffman-Taylor fingering. A Dynamical Solvability Scenario

A dynamical systems approach to competition of Saffman-Taylor fingers in a Hele-Shaw channel is developed. This is based on global analysis of the phase space flow of the low-dimensional ordinary-differential-equation sets associated with the classes of exact solutions of the problem without surface tension. Some simple examples are studied in detail. A general proof of the existence of finite-time singularities for broad classes of solutions is given. Solutions leading to finite-time interface pinchoff are also identified. The existence of a continuum of multifinger fixed points and its dynamical implications are discussed. We conclude that exact zero-surface tension solutions taken in a global sense as families of trajectories in phase space are unphysical because the multifinger fixed points are nonhyperbolic, and an unfolding does not exist within the same class of solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed points is argued to be essential to the physically correct qualitative description of finger competition. The restoring of hyperbolicity by surface tension is proposed as the key point to formulate a generic dynamical solvability scenario for interfacial pattern selection.