Willi-Hans Steeb

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 formulas for the ponderomotive force in a traveling wave of arbitrary strength are presented. It is further shown that the pondermotive 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.

Transition to hyperchaos in coupled generalized van der Pol equations

Abstract It has been shown that two coupled generalized van der Pol equations can show behaviour, i.e. the first two one-dimensional Lyapunov exponents are positive. The scaling law for transition from chaos to hyperchaos based on the properties of the Poincare map has been found. For fixed parameter values we also found that different behaviours of the system, such as limit cycles, chaos and hyperchaos, can coexist.

Finite State Machines

Finite state machines [49, 67] provide a visual representation of algorithms. Algorithms are implemented on a machine with a finite number of states representing the state of the algorithm. This provides an abstract way of designing algorithms. The chapter will only cover deterministic machines (the actions of the machines are determined uniquely).

Sequential clustering: tracking down the most natural clusters

Sequential superparamagnetic clustering (SSC) is a substantial extension of the superparamagnetic clustering approach (SC). We demonstrate that the novel method is able to master the important problem of inhomogeneous classes in the feature space. By fully exploiting the non-parametric properties of SC, the method is able to find the natural clusters even if they are highly different in shape and density. In such situations, concurrent methods normally fail. We present the results from a fully automated implementation of SSC (applications to chemical data and visual scene analysis) and provide analytical evidence of why the method works.

Generalized liouville equation, entropy, and dynamic systems containing limit cycles

The relation between the generalized Liouville equation, entropy production rates and autonomous systems of differential equations containing limit cycles is investigated. Moreover, the connection between the generalized Liouville equation and the Lie derivative of a differential form with respect to a vector field is discussed.

Painleve analysis for a time-dependent Kadomtsev-Petviashvili equation

Abstract We demonstrate that a time-dependent Kadomtsev-Petviashvili equation has the Painleve property for partial differential equations. Truncating the Painleve expansion yields an auto Backlund transformation and a representation in Zakharov-Shabat form.

Chaotic behaviour and limit cycle behaviour of anharmonic systems with periodic external perturbations

The chaotic behaviour and limit cycle behviaiour of the dynamical system x + αx + dV(x)dx = ƒ cos(Ωt) is investigated for various potentials V and parameters values α, ƒ and Ω.

Avoided level crossings and Riemann sheet structure

The Riemann sheet structure of the energy levels En(λ) of an N‐dimensional symmetric matrix problem of the form H0+λH1 is discussed. It is shown that the singularities of the energy levels in the complex λ plane are related to avoided level crossings. It is argued that locally the sheet structure is like that of a two‐, three‐, or four‐dimensional problem as far as two, three, or four adjacent levels are concerned. Expressions are given for adjacent levels displaying explicitly the Riemann sheet structure on a semiglobal footing.

Projective Noise Cleaning With Dynamic Neighborhood Selection

In recent years, several methods of noise cleaning have been devised, of which projective methods have been particularly effective. In our paper, we explain in detail why orthogonal projections are nonoptimal and how the nonorthogonal projections suggested by Grassberger et al., naturally emerge from the SVD method. We show that this approach when combined with a dynamic neighborhood selection yields optimal results of noise cleaning.

Chaos in limit cycle systems with external periodic excitations

Abstract Various limit cycle systems in the plane with an external periodic excitation are investigated from the point of view of chaotic behavior. Numerical studies are performed and a singular point analysis is carried out.