R. S. Gioggia

Calculating the dimension of attractors from small data sets

Abstract Calculations of the order-2 information dimension for data sets generated from numerical simulations as well as from an experiment show that it is possible to distinguish between periodic, chaotic and random signals, as well as to characterize different kinds of chaotic behavior using relatively small, noisy data sets.

Lasers and Brains: Complex Systems with Low-Dimensional Attractors

The quantification of complex dynamical phenomena associated with motions on strange attractors has made available a tool of considerable power for the analysis of systems which display aperiodic or apparently random temporal behavior. Until recently, aperiodic phenomena were described primarily in terms of snapshots of time sequences, power spectra, or correlation functions. These made possible some qualitative or pictorial analyses but did not provide simple numerical criteria suitable for more quantitative studies. During the past decade, spectral studies have made possible the identification of a few characteristic routes to apparently chaotic behavior in hydrodynamic[l–3], chemical [4], optical [5–7, 16, 29], and electronic [8, 9] systems. There were very strong indications that the complex motions, characterized by broadband spectra, to which these routes led, were in fact motions on strange attractors.

Definitions Of Chaos And Measuring Its Characteristics

The topic of this symposium is Optical Chaos. It is essential for those working in this field to know how to recognize, identify and characterize chaotic behavior and how to distinguish it from noise. General principles, definitions of terminology, and some brief review and commentary on the current status of techniques for measuring chaos are presented.

Lamb-dip-like behavior in each spectral component of a spontaneously pulsing, single-mode, standing-wave gas laser

The optical spectra corresponding to complex intensity pulsation patterns near line-center tuning of single-mode, standing-wave gas lasers contain some simple indicators of the origins of their complexity. Heterodyne measurements show that on detuning of the laser the power density in each component of the optical spectrum undergoes a Lamb-dip-like reduction as it passes through resonance with the zero-velocity atoms. These observations help to explain the sensitive dependence of the dynamics of standing-wave gas lasers on detuning within the Lamp dip and distortions of the Lamb-dip shapes when the lasers are unstable.

Experimental Studies of Instabilities and Chaos in Single-Mode, Inhomogeneously Broadened Gas Lasers

Inhomogeneously broadened lasers offer a complex problem for detailed theoretical modelling, particularly when the broadening arises from Doppler shifts of the resonant frequencies because of the thermal spread of velocities of individual atoms, as is true for gas lasers. Nevertheless there are several high-gain transitions in gas lasers (3.3 9 μm in neon, and 3.51 μm and 5.57 μm in xenon) which are particularly easy to use in constructing laboratory lasers. We are fortunate that, despite the formal difficulties, these lasers often lend themselves to straight-forward pictorial interpretations in terms of gain and dispersion for many steady-state (and some simple time-dependent) behaviors in the interaction between fields and the material. This makes qualitative discussions of the physics of these devices not only possible but instructive1–9.

Chaotic Dynamical Behavior in Lasers

We describe new uses of the spectrum of the output of a laser system to determine details of the changes in the dynamical behavior of lasers exhibiting highly nonlinear dynamics including deterministic chaos.