Nonlinear dynamical systems and classical orthogonal polynomials

It is demonstrated that nonlinear dynamical systems with analytic nonlinearities can be brought down to the abstract Schr\"odinger equation in Hilbert space with boson Hamiltonian. The Fourier coefficients of the expansion of solutions to the Schr\"odinger equation in the particular occupation number representation are expressed by means of the classical orthogonal polynomials. The introduced formalism amounts a generalization of the classical methods for linearization of nonlinear differential equations such as the Carleman embedding technique and Koopman approach.Comment: 21 pages latex, uses revte

Chaotic saddles in nonlinear modulational interactions in a plasma

A nonlinear model of modulational processes in the subsonic regime involving a linearly unstable wave and two linearly damped waves with different damping rates in a plasma is studied numerically. We compute the maximum Lyapunov exponent as a function of the damping rates in a two-parameter space, and identify shrimp-shaped self-similar structures in the parameter space. By varying the damping rate of the low-frequency wave, we construct bifurcation diagrams and focus on a saddle-node bifurcation and an interior crisis associated with a periodic window. We detect chaotic saddles and their stable and unstable manifolds, and demonstrate how the connection between two chaotic saddles via coupling unstable periodic orbits can result in a crisis-induced intermittency. The relevance of this work for the understanding of modulational processes observed in plasmas and fluids is discussed.Comment: Physics of Plasmas, in pres

Non-linear autonomous systems of differential equations and Carleman linearization procedure

AbstractThe non-linear autonomous of differential equations ẋi=∑jaijxj+∑j,kbijkxjxk(ẋi=dxi/dt, i, j, k= 1,2,…n) which plays an important role in chemical kinetics and other fields of physics (turbulence and plasma physics) is investigated using the Carleman linearization procedure

Relativistic ponderomotive force, uphill acceleration, and transition to chaos

Starting from a covariant cycle-averaged Lagrangian the relativistic oscillation center equation of motion of a point charge is deduced and analytical formulae for the ponderomotive force in a travelling wave of arbitrary strength are presented. It is further shown that the ponderomotive forces for transverse and longitudinal waves are different; in the latter, uphill acceleration can occur. In a standing wave there exists a threshold intensity above which, owing to transition to chaos, the secular motion can no longer be described by a regular ponderomotive force. PACS number(s): 52.20.Dq,05.45.+b,52.35.Mw,52.60.+hComment: 8 pages, RevTeX, 3 figures in PostScript, see also http://www.physik.th-darmstadt.de/tqe

Comment on the Shiner-Davison-Landsberg Measure

The complexity measure from Shiner et al. [Physical Review E 59, 1999, 1459-1464] (henceforth abbreviated as SDL-measure) has recently been the subject of a fierce debate. We discuss the properties and shortcomings of this measure, from the point of view of our recently constructed fundamental, statistical mechanics-based measures of complexity Cs(γ,β) [Stoop et al., J. Stat. Phys. 114, 2004, 1127-1137]. We show explicitly, what the shortcomings of the SDL-measure are: It is over-universal, and the implemented temperature dependence is trivial. We also show how the original SDL-approach can be modified to rule out these points of critique. Results of this modification are shown for the logistic parabol

SHynergie: Development of a virtual project laboratory for monitoring hydraulic stimulations

Hydraulic stimulations are the primary means of developing subsurface reservoirs regarding the extent of fluid transport in them. The associated creation or conditioning of a system of hydraulic conduits involves a range of hydraulic and mechanical processes but also chemical reactions, such as dissolution and precipitation, may affect the stimulation result on time scales as short as hours. In the light of the extent and complexity of these processes, the steering potential for the operator of a stimulation critically depends on the ability to integrate the maximum amount of site-specific information with profound process understanding and a large spectrum of experience. We report on the development of a virtual project laboratory for monitoring hydraulic stimulations within the project SHynergie (http://www.ruhr-uni-bochum.de/shynergie/). The concept of the laboratory envisioned product that constitutes a preparing and accompanying rather than post-processing instrument ultimately accessible to persons responsible for a project over a web-repository. The virtual laboratory consists of a data base, a toolbox, and a model-building environment. Entries in the data base are of two categories. On the one hand, selected mineral and rock properties are provided from the literature. On the other hand, project-specific entries of any format can be made that are assigned attributes regarding their use in a stimulation problem at hand. The toolbox is interactive and allows the user to perform calculations of effective properties and simulations of different types (e.g., wave propagation in a reservoir, hydraulic test). The model component is also hybrid. The laboratory provides a library of models reflecting a range of scenarios but also allows the user to develop a site-specific model constituting the basis for simulations. The laboratory offers the option to use its components following the typical workflow of a stimulation project. The toolbox incorporates simulation instruments developed in the course of the SHynergie project that account for the experimental and modeling results of the various sub-projects

Localization and litigation of radio frequency interference for interferometric arrays

Thesis (PhD)--Stellenbosch University, 2018.ENGLISH ABSTRACT: Radio telescopes have increased exponentially in sensitivity ever since the first single dish radio telescopes were built in the 1930's. This trend continues with the development of next generation telescopes such as the Square Kilometre Array (SKA). Parallel to the development of radio telescopes, has been the rapid expansion of telecommunication technologies. Consequently, radio telescopes are becoming more sensitive in an environment with ever increasing radio frequency interference (RFI). The ideal solution to RFI that is detected by a radio telescope is to locate its source and then have it removed. Removal of the source is usually only possible if it is occurring in a protected band or the radio telescope is in a radio quiet zone. Unfortunately, most of the radio spectrum has been allocated to active communication services and not all radio telescopes are in radio quiet zones. The alternative is to mitigate its effect using methods such as spatial RFI mitigation. The contributions of this PhD dissertation are twofold: firstly, a source localization algorithm that takes into account the constraints and advantages of the arrays used for radio astronomy has been developed; and secondly, existing spatial RFI mitigation techniques have been adapted to take into account the bandwidth of the RFI signals. The computationally efficient localization algorithm that was developed is best suited for interferometric arrays with low array beam sidelobes. Two variants of the algorithm were developed, one that works for sources in the near-field and the other for far-field sources. In the near-field, the computational complexity of the algorithm is linear with search grid size compared to cubic scaling of the state-of-the-art 3-D MUSIC method. The trade-off is that the proposed algorithm requires a once-off a priori calculation and storing of weighting matrices. In an experiment using a station of the Low Frequency Array (LOFAR) a hexacopter was flown around the array, at a mean radial distance of 190 m, broadcasting a signal. The mean error in distance between the estimated position of the hexacopter and the GPS position of the hexacopter was 2 m for a wavelength of 6.7 m. The non-narrowband RFI mitigation method developed consists of a second order filter that is used to mitigate powerful RFI with bandwidth sufficient to cause aberrations that are below the noise, but with power that competes with the astronomical sources. The second order filter consists of a first order subspace subtraction filter combined with a flat frequency response model for the RFI source. Taking into account mutual coupling as well as a calibration step to account for unknown complex gains, the algorithm was found to process approximately 1.6 times more bandwidth than using just a first order subspace subtraction filter.AFRIKAANSE OPSOMMING: Sedert die eerste enkelskottel radioteleskope in die 1930's gebou is, het die sensitiwiteit van radioteleskope eksponensieël toegeneem. Hierdie tendens gaan voort met die ontwikkeling van volgende generasie teleskope, soos byvoorbeeld die Square Kilometer Array (SKA). In parallel met die ontwikkeling van radioteleskope, het telekommunikasietegnologieë ook vinnig uitgebrei. Gevolglik word radioteleskope meer sensitief in 'n omgewing met toenemende radiofrekwensie-inmenging (RFI). Die optimale oplossing vir RFI is om die bron daarvan op te spoor en te verwyder. Verwydering van die bron is gewoonlik net moontlik as dit teenwoording is in 'n beskermde band of as die radio teleskoop in 'n radio-stil gebied is. Ongelukkig is meeste van die radio spectrum toegeken aan kommunikasiedienste en nie alle radio teleskope is in radio-stil gebiede nie. Die alternatief om die effek daarvan te mitigeer deur middel van metodes soos ruimtelike RFI-mitigasie. Die bydraes van hierdie doktorale proefskrif is tweeledig: eerstens, die ontwikkeling van 'n bronlokaliseringsalgoritme wat die beperkings en voordele van die skikkings wat gebruik word vir radio astronomie in ag neem en tweedens, die aanpassing van bestaande ruimtelike RFI mitigeringstegnieke om die bandwydte van die RFI seine in ag te neem. Die berekeningsdoeltreffende lokaliseringsalgoritme wat ontwikkel is, is die beste geskik vir interferometriese skikkings met lae samestelling-bundel sylobbe. Twee weergawes van die algoritme is ontwikkel, die eerste hanteer bronne in die nabyveld en die ander hanteer vêrveld bronne. In die nabyveld is die berekeningskompleksiteit van die algoritme lineêr met soektogroostergrootte in vergelyking met die kubieke skalering van die 3-D MUSIC-metode. Die nadeel is dat die voorgestelde algoritme 'n eenmalige a priori berekening en stoor van gewigsmatrikse vereis. In 'n eksperiment by 'n stasie van die Low Frequency Array (LOFAR), het 'n heksakopter oor die skikking gevlieg met 'n gemiddelde radiale afstand van 190 m en 'n sein uitgesaai. Die gemiddelde fout in die afstand tussen die beraamde posisie van die heksakopter en die GPS-posisie van die heksakopter was 2 m vir 'n golflengte van 6.7 m. Die nie-smalband RFI mitigasie metode wat ontwikkel is, fasiliteer die de-finieering van 'n tweede-orde filter wat gebruik word om kragtige RFI met bandwydte verwante krag onder die geruis, maar met krag wat met die astronomiese bronne kompeteer, te mitigeer. Die tweede order filter bestaan uit 'n eerste orde subruimte verminderingsfilter gekombineer met 'n plat frekwensie responsmodel vir die RFI bron. Met inagneming van wedersydse koppeling asook 'n kalibrasie stap om vir onbekende komplekse antenna aanwinste voorsiening te maak, is gevind dat die algoritme ongeveer 1.6 meer bandwydte kan verwerk as 'n eerste orde subruimte verminderingsfiter

The converse problem for the multipotentialisation of evolution equations and systems

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the {\it converse problem}. Although we mainly study a method for $(1+1)$-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-B\"acklund transformations for the equations/systems. In this respect we define the {\it triangular-auto-B\"acklund transformation} and derive its connections to the converse problem. Several explicit examples are given. In particular we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.Comment: 31 Pages, 7 diagrams, submitted for consideratio

A Geometrical Method of Decoupling

The computation of tunes and matched beam distributions are essential steps in the analysis of circular accelerators. If certain symmetries - like midplane symmetrie - are present, then it is possible to treat the betatron motion in the horizontal, the vertical plane and (under certain circumstances) the longitudinal motion separately using the well-known Courant-Snyder theory, or to apply transformations that have been described previously as for instance the method of Teng and Edwards. In a preceeding paper it has been shown that this method requires a modification for the treatment of isochronous cyclotrons with non-negligible space charge forces. Unfortunately the modification was numerically not as stable as desired and it was still unclear, if the extension would work for all thinkable cases. Hence a systematic derivation of a more general treatment seemed advisable. In a second paper the author suggested the use of real Dirac matrices as basic tools to coupled linear optics and gave a straightforward recipe to decouple positive definite Hamiltonians with imaginary eigenvalues. In this article this method is generalized and simplified in order to formulate a straightforward method to decouple Hamiltonian matrices with eigenvalues on the real and the imaginary axis. It is shown that this algebraic decoupling is closely related to a geometric "decoupling" by the orthogonalization of the vectors $\vec E$, $\vec B$ and $\vec P$, that were introduced with the so-called "electromechanical equivalence". We present a structure-preserving block-diagonalization of symplectic or Hamiltonian matrices, respectively. When used iteratively, the decoupling algorithm can also be applied to n-dimensional systems and requires ${\cal O}(n^2)$ iterations to converge to a given precision.Comment: 13 pages, 1 figur

Painlev\'{e} test of coupled Gross-Pitaevskii equations

Painlev\'{e} test of the coupled Gross-Pitaevskii equations has been carried out with the result that the coupled equations pass the P-test only if a special relation containing system parameters (masses, scattering lengths) is satisfied. Computer algebra is applied to evaluate j=4 compatibility condition for admissible external potentials. Appearance of an arbitrary real potential embedded in the external potentials is shown to be the consequence of the coupling. Connection with recent experiments related to stability of two-component Bose-Einstein condensates of Rb atoms is discussed.Comment: 13 pages, no figure