Peter V. E. McClintock

Noise in Nonlinear Dynamical Systems

Abstract Noise is commonly regarded as having a destructive but relatively innocuous effect, blurring our view of a system but having no effect on the underlying processes involved. In this paper we show, using examples from stochastic nonlinear dynamics, that these intuitive ideas about noise can be very misleading. For example, an effect known as stochastic resonance means that the addition of extra noise to a system can actually improve the signal-to-noise ratio.

Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium.

We report observation of an inverse energy cascade in second sound acoustic turbulence in He II. Its onset occurs above a critical driving energy and it is accompanied by giant waves that constitute an acoustic analogue of the rogue waves that occasionally appear on the surface of the ocean. The theory of the phenomenon is developed and shown to be in good agreement with the experiments.

Analogue studies of nonlinear systems

The design of analogue electronic experiments to investigate phenomena in nonlinear dynamics, especially stochastic phenomena, is described in practical terms. The advantages and disadvantages of this approach, in comparison to more conventional digital methods, are discussed. It is pointed out that analogue simulation provides a simple, inexpensive, technique that is easily applied in any laboratory to facilitate the design and implementation of complicated and expensive experimental projects; and that there are some important problems for which analogue methods have so far provided the only experimental approach. Applications to several topical problems are reviewed. Large rare fluctuations are studied through measurements of the prehistory probability distribution, thereby testing for the first time some fundamental tenets of fluctuation theory. It has thus been shown for example that, whereas the fluctuations of equilibrium systems obey time-reversal symmetry, those under non-equilibrium conditions are temporally asymmetric. Stochastic resonance, in which the signal-to-noise ratio for a weak periodic signal in a nonlinear system can be enhanced by added noise, has been widely studied by analogue methods, and the main results are reviewed; the closely related phenomena of noise-enhanced heterodyning and noise-induced linearization are also described. Selected examples of the use of analogue methods for the study of transient phenomena in time-evolving systems are reviewed. Analogue experiments with quasimonochromatic noise, whose power spectral density is peaked at some characteristic frequency, have led to the discovery of a range of interesting and often counter-intuitive effects. These are reviewed and related to large fluctuation phenomena. Analogue studies of two examples of deterministic nonlinear effects, modulation-induced negative differential resistance (MINDR) and zero-dispersion nonlinear resonance (ZDNR) are described. Finally, some speculative remarks about possible future directions and applications of analogue experiments are discussed.

Stochastic resonance in perspective.

SummaryWe outline the historical development of stochastic resonance (SR), a phenomenon in which the signal and/or the signal-to-noise ratio in a nonlinear system increase with increasing intensity of noise. We discuss basic theoretical ideas explaining and describing SR, and we review some revealing experimental data that place SR within the wider context of statistical physics. We emphasize the close relationship of SR to some effects that are well known in condensed-matter physics.

Wavelet Phase Coherence Analysis: Application to Skin Temperature and Blood Flow

The technique of wavelet phase coherence analysis is introduced and used to explore relationships between oscillations on blood flow and temperature in the skin of 10 healthy subjects. Their skin temperature and blood flow were continuously recorded: under basal conditions for 30 min; during local cooling of the skin with an ice-pack for 20 min: and 30 min thereafter. The group mean basal skin temperature of 33.4°C was decreased to 29.2°C during the cooling period, and had recovered to 32.1°C by the end of the recording. The wavelet transform was used to obtain the time–frequency content of the two signals, and their coherence. It is shown that cooling increases coherence to a statistically significant extent in two frequency intervals, around 0.007 and 0.1 Hz, suggesting that these oscillatory components are involved in the regulation of skin temperature when cold is applied as a stress.

Irreversibility of classical fluctuations studied in analogue electrical circuits

Fluctuations around some average or equilibrium state arise universally in physical systems. Large fluctuations — fluctuations that are much larger than average — occur only rarely but are responsible for many physical processes, such as nucleation in phase transitions, chemical reactions, mutations in DNA sequences, protein transport in cells and failure of electronic devices. They lie at the heart of many discussions of how the irreversible thermodynamic behaviour of bulk matter relates to the reversible (classical or quantum-mechanical) laws describing the constituent atoms and molecules. Large fluctuations can be described theoretically using hamiltonian, and equivalent path-integral formulations, but these approaches remain largely untested experimentally, mainly because such fluctuations are rare and also because only recently was an appropriate statistical distribution function formulated. It was shown recently, however, that experiments on fluctuations using analogue electronic circuits allow the phase-space trajectories of fluctuations in a dynamical system to be observed directly. Here we show that this approach can be used to identify a fundamental distinction between two types of random motion: fluctuational motion, which takes the system away from a stable state, and relaxational motion back towards this state. We suggest that macroscopic irreversibility is related to temporal asymmetry of these two types of motion, which in turn implies a lack of detailed balance and corresponds to non-differentiability of the generalized nonequilibrium potential in which the motion takes place.

STOCHASTIC RESONANCE IN MONOSTABLE SYSTEMS

The first observations of noise-induced enhancements and phase shifts of a weak periodic signal-characteristics signatures of stochastic resonance (SR)-are reported for a monostable system. The results are shown to be in good agreement with a theoretical description based on linear-response theory and the fluctuation dissipation theorem. It is argued that SR is a general phenomenon that can in principle occur for any underdamped nonlinear oscillator.

Interactions between cardiac, respiratory and EEG-δ oscillations in rats during anaesthesia

We hypothesized that, associated with the state of anaesthesia, characteristic changes exist in both cardio‐respiratory and cerebral oscillator parameters and couplings, perhaps varying with depth of anaesthesia. Electrocardiograms (ECGs), respiration and electroencephalograms (EEGs) were recorded from two groups of 10 rats during the entire course of anaesthesia following the administration of a single bolus of ketamine–xylazine (KX group) or pentobarbital (PB group). The phase dynamics approach was then used to extract the instantaneous frequencies of heart beat, respiration and slow δ‐waves (within 0.5–3.5 Hz). The amplitudes of δ‐ and θ‐waves were analysed by use of a time–frequency representation of the EEG signal within 0.5–7.5 Hz obtained by wavelet transformation, using the Morlet mother wavelet. For the KX group, where slow δ‐waves constituted the dominant spectral component, the Hilbert transform was applied to obtain the instantaneous δ‐frequency. The θ‐activity was spread over too wide a spectral range for its phase to be meaningfully defined. For both agents, we observed two distinct phases of anaesthesia, with a marked increase in θ‐wave activity occurring on passage from a deeper phase of anaesthesia to a shallower one. In other respects, the effects of the two anaesthetics were very different. For KX anaesthesia, the two phases were separated by a marked change in all three instantaneous frequencies: stable, deep, anaesthesia with small frequency variability was followed by a sharp transition to shallow anaesthesia with large frequency variability, lasting until the animal awoke. The transition occurred 16–76 min after injection of the anaesthetic, with simultaneous reduction in the δ‐wave amplitude. For PB anaesthesia, the two epochs were separated by the return of a positive response to the pinch test at 53–94 min, following which it took a further period of 45–70 min for the animal to awaken. δ‐Waves were not apparent at any stage of PB anaesthesia. We applied non‐linear dynamics and information theory to seek evidence of causal relationships between the cardiac, respiratory and slow δ‐oscillations. We demonstrate that, for both groups, respiration drives the cardiac oscillator during deep anaesthesia. During shallow KX anaesthesia the direction either reverses, or the cardio‐respiratory interaction becomes insignificant; in the deep phase, there is a unidirectional deterministic interaction of respiration with slow δ‐oscillations. For PB anaesthesia, the cardio‐respiratory interaction weakens during the second phase but, otherwise, there is no observable change in the interactions. We conclude that non‐linear dynamics and information theory can be used to identify different stages of anaesthesia and the effects of different anaesthetics.

Stochastic resonance in electrical circuits. II. Nonconventional stochastic resonance

For pt.I see ibid., vol.46, no.9, pp.1205-14 (1999). Stochastic resonance (SR), in which a periodic signal in a nonlinear system can be amplified by added noise, is discussed. The application of circuit modeling techniques to the conventional form of SR, which occurs in static bistable potentials, was considered in a companion paper. Here, the investigation of nonconventional forms of SR in part using similar electronic techniques is described. In the small-signal limit, the results are well described in terms of linear response theory. Some other phenomena of topical interest, closely related to SR, are also treated.

Changes in the dynamical behavior of nonlinear systems induced by noise

Abstract Weak noise acting upon a nonlinear dynamical system can have far-reaching consequences. The fundamental underlying problem – that of large deviations of a nonlinear system away from a stable or metastable state, sometimes resulting in a transition to a new stationary state, in response to weak additive or multiplicative noise – has long attracted the attention of physicists. This is partly because of its wide applicability, and partly because it bears on the origins of temporal irreversibility in physical processes. During the last few years it has become apparent that, in a system far from thermal equilibrium, even small noise can also result in qualitative change in the systems properties, e.g., the transformation of an unstable equilibrium state into a stable one, and vice versa, the occurrence of multistability and multimodality, the appearance of a mean field, the excitation of noise-induced oscillations, and noise-induced transport (stochastic ratchets). A representative selection of such phenomena is discussed and analyzed, and recent progress made towards their understanding is reviewed.