Attractor Regime Research Articles

Number of degrees of freedom of a lip-reed in brass-instrument playing

A lip-reed is a double continuous system that potentially exhibits several degrees of freedom. Numerical simulations for sound synthesis as well as fitting a model with measurements naturally improve while increasing the number of degrees of freedom that are put into the model. However, this parameter is worth investigating as such. In a first approach, lips are coupled to a Helmholtz resonator whose resonance factor is actively reinforced, as described in Chen and Weinreich [‘‘Nature of lip reed,’’ J. Acoust. Soc. Am. 99, 1227–1233 (1996)]. The dimensionality of the attractor or permanent regime of this nonlinear dynamical system is computed with embedding techniques of the corresponding time-series and information-based measurements [Gershenfeld et al., Time Series Prediction (Addison-Wesley, Reading, MA, 1994)]. Preliminary results indicate a dimension of 1, 2 or 3 for different couplings between the lips and the resonator. The link with the number of freedom of the lip-reed will be discussed. [Work supported in part by CNRS and NSF.]

Passive particle dynamics in a flow exhibiting transition to turbulence

The behavior of a passive particle in a flow that exhibits bifurcations in the transition to a turbulent regime is investigated. The flow considered is a variant of the Charney–Hasegawa–Mima equation. The scalar particle dynamics is considered for different regimes of the main flow. A regime of anomalous diffusion (hypodiffusion) is observed when the field has few harmonics whereas normal diffusion occurs in the strange attractor regime. The analysis of the singular orbit reveals the presence of traps and flights that control the transport.

An attractor regime for pulsed NMR in3He-B

We have found a new long-lived mode that can be generated in the regime of pulsed NMR in3He-B. The mode has a frequency different from both the Larmor one and that which was generated in the experiment of Corruccini and Osheroff by rf pulses longer than 104°. It does not correspond to stationary solutions of the Leggett-Takagi equations. The mode is due to the existence of an attractor that is a result of an interplay between nonlinear and dissipative effects in the spin dynamics of superfluid3He-B.

Statistical properties in the chaotic regime of the Maxwell-Bloch equations

The statistical properties of the electric field solution of the Maxwell-Bloch equations describing a single mode, homogeneously broadened laser in the mean field limit are investigated in the strange attractor regime. The electric field distribution sis calculated and it is found that the low order intensity moments are somewhat higher than those for thermal-chaotic light whilst the higher order moments are substantially lower. The field and intensity correlation functions are also calculated and found to exhibit radically different behaviour. The results are interpreted in terms of iterative map which is dederived using multiple time-scale perturbation theory. It is shown that a simple random phasor model is compatible with the numerical data.

SHOULD A RELIABLE INFORMATION PROCESSOR BE CHAOTIC?

Brain‐like structures have evolved by performing signal processing initially by minimizing “tracking errors” on a competitive basis. Such systems are highly complex and at the same time notoriously “disordered”. The functional trace of the cerebral cortex of the (human) brain is a good example. The Electroencephalogram (E.E.G) appears particularly fragmented during the execution of mental tasks, as well as during the recurrent episodes of R.E.M. sleep. A stochastically regular or a highly synchronized E.E.G on the other hand, characterises a drowsy (relaxing) or epileptic subject respectively and indicates—in both cases—a very incompetent information processor. We suggest that such behavioral changeovers are produced via bifurcations which trigger the thalamocortical non‐linear pacemaking oscillator to switch from an unstable limit cycle to a strange attractor regime (i.e. to chaos), or vice versa. Our analysis aims to show that the E.E.G's characteristics are not accidental but inevitable and even necessary and, therefore, functionally significant.