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

Internal erosion of granular materials – Identification of erodible fine particles as a basis for numerical calculations

In geohydromechanics internal erosion is a process which is still hardly to be quantified both spatially as well as temporally. The transport of fine particles, which is caused by increased hydraulic gradients, is influenced by the pore structure of the coarse grained fabric. The microstructural information of the pore constriction size distribution (CSD) of the solid skeleton has therefore to be taken into account when internal erosion is analyzed either analytically or numerically. The CSD geometrically defines the amount of fine particles, which potentially can be eroded away for a given hydraulic force. The contribution introduces experimental and numerical calculations which aim at the quantification of the amount of erodible fines. Based on this approach a multiphase continuum-based numerical model is used to back calculate the process of internal erosion for one material of the well-known experimental investigation of Skempton & Brogan (1994)[1]

Visualizing particle networks in granular media by in situ X-ray computed tomography

In this contribution, cylindrical samples consisting of monodisperse soft (rubber) and stiff (glass) particles are pre-stressed under uniaxial compression. Acoustic P-waves at ultrasound frequencies are superimposed into prepared samples with different soft-stiff volume fractions. Earlier investigations showed the importance of particles networks, i.e. force chains, in controlling the effective mechanical properties of particulate systems. Measured P-wave modulus showed a significant decline while more soft particles are added due to a change in microstructure. However, for small contents of soft particles, it could be observed that the P-wave modulus is increasing. For the understanding of such kinds of effects, detailed insight into the microstructure of the system is required. To gain this information and link it to the effective properties, we made use of high-resolution micro X-ray Computed Tomography (micro-XRCT) imaging and combined it with the classical stiffness characterization. Both performed in situ meaning inside the laboratory-based XRCT scanner. With micro-XRCT imaging, the granular microstructure can be visualized in 3d and characterized subsequently. By post-processing of the data, the individual grains of the particulate systems could be uniquely identified. Finally, the contact network of the packings which connects the center of particles was established to demonstrate the network transition from stiff- to soft-dominated regimes. This has allowed for unprecedented observations and a renewed understanding of particulate systems. It has been demonstrated that micro-XRCT scans of particles packings can be analyzed and compared in 3d to gain extensive information on the scale of the single particles. Here, the in situ setup and workflow from the start of acquiring images in situ till the post-processing of the image data is explained and demonstrated by selected results.Comment: 17 pages, 10 figures, submitted to Survey for Applied Mathematics and Mechanics (GAMM Mitteilungen

Kinematically Extended Continuum Theories: Correlation Between Microscopical Deformation and Macroscopical Strain Measures

The present work investigates the correlation between macrocscopical deformation modes and microscopical deformation modes. Thereby, the macroscopical deformation is represented by the strain-like quantities of the according macroscopical continuum theory while the microscopical deformation is expressed in the form of a Taylor series expansion. The use of an energy criterion makes it possible to derive a quantitative relation between microscopical and macroscopical deformation. The procedure is applied to different kinematically extended continuum theories on the macroscopical level. The investigation may help to select an optimal macroscopical continuum theory instead of choosing a theory based on phenomenological observations, whereby the optimal theory ist that one, which reflects the microscopical deformation behaviour best. The microscopical deformation behaviour depends on the topology of the microstructure under consideration. Thus, the optimal theory is affected by the topology of the microstructure

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