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

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

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

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

Lyapunov exponent and natural invariant density determination of chaotic maps: An iterative maximum entropy ansatz

We apply the maximum entropy principle to construct the natural invariant density and Lyapunov exponent of one-dimensional chaotic maps. Using a novel function reconstruction technique that is based on the solution of Hausdorff moment problem via maximizing Shannon entropy, we estimate the invariant density and the Lyapunov exponent of nonlinear maps in one-dimension from a knowledge of finite number of moments. The accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available. Furthermore, we also consider a very complex example for which no exact analytical result for invariant density is available. A comparison of our results to those available in the literature is also discussed.Comment: 16 pages including 6 figure