Nonlinear dynamics of delay systems: an overview.

Time delays play an important role in many fields such as engineering, physics or biology. Delays occur due to finite velocities of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delay systems can be described by delay differential equations and often include non-negligible nonlinear effects. This overview article introduces the theme issue 'Nonlinear dynamics of delay systems', which contains new fundamental results in this interdisciplinary field as well as recent developments in applications. Fundamentally, new results were obtained especially for systems with time-varying delay and state-dependent delay and for delay system with noise, which do often appear in real systems in engineering and nature. The applications range from climate modelling over network dynamics and laser systems with feedback to human balancing and machine tool chatter. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Resonant Doppler effect in systems with variable delay.

We demonstrate that a time-varying delay in nonlinear systems leads to a rich variety of dynamical behaviour, which cannot be observed in systems with constant delay. We show that the effect of the delay variation is similar to the Doppler effect with self-feedback. We distinguish between the non-resonant and the resonant Doppler effect corresponding to the dichotomy between conservative delays and dissipative delays. The non-resonant Doppler effect leads to a quasi-periodic frequency modulation of the signal, but the qualitative properties of the solution are the same as for constant delays. By contrast, the resonant Doppler effect leads to fundamentally different solutions characterized by low- and high-frequency phases with a clear separation between them. This is equivalent to time-multiplexed dynamics and can be used to design systems with well-defined multistable solutions or temporal switching between different chaotic and periodic dynamics. We systematically study chaotic dynamics in systems with large dissipative delay, which we call generalized laminar chaos. We derive a criterion for the occurrence of different orders of generalized laminar chaos, where the order is related to the dimension of the chaotic attractor. The recently found laminar chaos with constant plateaus in the low-frequency phases is the zeroth-order case with a very low dimension compared to the known high dimension of turbulent chaos in systems with conservative delay. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Ripple formation through an interface instability from moving growth and erosion sources

The propagation of material interfaces is investigated under the action of a localized moving source which deposits or removes material. Among others the latter process applies to beam cutting techniques. We develop a Kuramoto-Sivashinsky-type model and find a new type of ripple forming mechanism. This theory offers a new explanation for the occurrence of striation patterns which often degrade the quality of cutting edges.

Self-averaging of an order parameter in randomly coupled limit-cycle oscillators

In our recent paper [Phys. Rev. E 58, 1789 (1998)] we found notable deviations from a power-law decay for the "magnetization" of a system of coupled phase oscillators with random interactions claimed by Daido in Phys. Rev. Lett. 68, 1072 (1992). For another long-time property, the Lyaponov exponent, we found that his numerical procedure showed strong time discretization effects and we suspected a similar effect for the algebraic decay. In the Comment to our paper [preceding paper, Phys. Rev. E 61, 2145 (2000)] Daido made clear that the power law behavior was only claimed for the sample averaged magnetization [Z] and he presented new, more accurate numerical results which provide evidence for a power-law decay of this quantity. Our results, however, were obtained for Z itself and not for [Z]. In addition, we have taken the intrinsic oscillator frequencies as Gaussian random variables, while Daido in his new and apparently also in his earlier simulations used a deterministic approximation to the Gaussian distribution. Due to the differences in the observed quantity and the model assumptions our and Daido's results may be compatible.

Analysis, classification, and coding of multielectrode spike trains with hidden Markov models.

It is shown that hidden Markov models (HMMs) are a powerful tool in the analysis of multielectrode data. This is demonstrated for a 30-electrode measurement of neuronal spike activity in the monkey's visual cortex during the application of different visual stimuli. HMMs with optimized parameters code the information contained in the spatiotemporal discharge patterns as a probabilistic function of a Markov process and thus provide abstract dynamical models of the pattern-generating process. We compare HMMs obtained from vector-quantized data with models in which parametrized output processes such as multivariate Poisson or binomial distributions are assumed. In the latter cases the visual stimuli are recognized at rates of more than 90% from the neuronal spike patterns. An analysis of the models obtained reveals important aspects of the coding of information in the brain. For example, we identify relevant time scales and characterize the degree and nature of the spatiotemporal variations on these scales.

Free radicals from X-irradiated single crystals of uridine-5'-phosphate disodium salt.

X-irradiation of single crystals of uridine-5'-phosphate (disodium salt) between 10 and 300 K as well as storage of irradiated crystals at 300 K produces at least seven different radical species. Between 10 and 77 K, the uracil base anion and a secondary alkoxy radical at the ribose-O3'-site are formed. The latter transforms into a C5'-centred alkylphosphate species between 110 and 130 K which in turn decays between 180 and 220 K under formation of a base 5-yl hydrogen addition radical. Irradiation at 300 K additionally produces the base-located 6-yl radical together with a radical tentatively assigned to the doubly protonated base anion. Storage of crystals for several months results in decay of most of these species leaving a radical possibly located at c5' of the ribose. The spectral parameters of these radicals are given and discussed.