Julia sets for the gamma recursion in nonlinear psychophysics

Campbell, Edward A.; Gregson, Robert A. M.

doi:10.1007/bf00046909

Cited by 8 publications

(3 citation statements)

References 10 publications

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“…The r recursion (Campbell & Gregson, 1990;Gregson, 1988Gregson, , 1989aGregson, , 1989bGregson & Britton, 1989;Price & Gregson, 1988) in two-variable form is…”

Section: Attractor Propertiesmentioning

confidence: 99%

Similarities of low-dimensional chaotic auditory attractor sequences to quasirandom noise

Gregson

Harvey

1992

Perception & Psychophysics

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The production ofsequences ofsounds of various pitch levels from the algebra of chaotic attractors' trajectories is relatively straightforward. Meyer-Kress (cited in Kaneko, 1986) suggested that such sequences would be distinguishable from random independent identically distributed sequences. In psychophysical terms, this is a pattern-discrimination or pattern-similarity perception task, but these two tasks are not exactly the same thing. Nine attractors from the algebras of Henon, Zaslavskii (1978), Kaplan and Yorke (1979), Lorenz, and Gregson, and the logistic and Baker transformations, were paired with 10 realizations of a random series. The identification of the random member in each pair, the confidence of identification, and the perceived pairwise similarity were recorded by 65 subjects without initial feedback and by 76 subjects with initial feedback on five trials only, for each of 20 such pairs. The results indicate varying degress of discriminability; they can be expresssed in an analog ofthe receiver-operating characteristics of the attractors. There is no evidence of any homogeneous basis for the discrimination, and subjects who perform better are apparently not using the same bases as those who perform poorly. The fractal dimensionality of attractors may furnish a basis for their recognition, or the consequent autoregressive spectra induced in finite (short) samples, but recent work suggests the latter spectra can be insensitive to low-dimensional attractor dynamics.A dynamic process is by definition something that evolves through time, and its mathematical representation is characterized both by the variables that themselves evolve, creating a time series of realizations of the process (either continuously or in discrete steps), and by parameters that determine the form of the evolution. The values of these parameters are critical, because in a nonlinear process, very small changes in the parameters can induce vast changes in the evolutionary dynamics. There is no necessary proportionality between changes in the values of the parameters governing the evolution of the process in time and the changes in the evolutionary path itself. This evolution of the process is sometimes called the trajectory of the variables.In some circumstances, which can be specified completely by the parameter values and the starting values of the process variables, the trajectory locks into a stable pattern, which is called an attractor. An attractor can be a single point, a cycle of fixed periodicity in the time units of the trajectory, or can take one of various more complicated forms, some of which are called chaotic. Chaotic behavior resembles random behavior but is completely deterministic. It is not locally predictable beyond a very short time ahead, and the degree of its unpredictability is quantifiable from properties of its algebra. The Lyapunov exponents are the most usual way to define the dynamics in this sense.The Lyapunov exponents measure the rate at which nearby trajectories diverge; "nearby" implies that t...

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“…The r recursion (Campbell & Gregson, 1990;Gregson, 1988Gregson, , 1989aGregson, , 1989bGregson & Britton, 1989;Price & Gregson, 1988) in two-variable form is…”

Section: Attractor Propertiesmentioning

confidence: 99%

Similarities of low-dimensional chaotic auditory attractor sequences to quasirandom noise

Gregson

Harvey

1992

Perception & Psychophysics

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“…It is found that this recursion exhibits very complicated dynamics (Campbell & Gregson, 1990; Price & Gregson, 1988) with interplay of the effects of a and e. As ( 7 ) is a nonlinear mapping from a onto Y, for a given Yo, values of a which correspond to stimulus levels 4 (input does not map directly onto Y) can induce single or multiple values of Y,, though obviously not both at once, and the evolution of Y becomes periodic, quasi-periodic or chaotic as a function of the two parameters a and e. If we generate, in time, a sequence of u values as stimulus inputs, which is almost continuous and monotone increasing, the corresponding sequence of Y( Re) values, which predict the observable outputs, will not necessarily exhibit the same monotonicity. The r trajectory can be written as two evolutions, cross-coupled in two real variables (Campbell & Gregson, 1990; Gregson, 1991) but this is not to be interpreted as a signal and a noise channel in parallel; the e variable's main role is to modify critically the shape of the psychometric function, and the points at which there are changes in the system dynamics from point output (single-valued Y,(Ke)) to limit cycling within r trajectory evolution, and on to chaos. Chaos is only associated with sensory overload, which does not appear usually in the present task context.…”

Section: Nonlinear Analysismentioning

confidence: 99%

The psychophysical method of limits: What actually happens in a nonlinear context?

Gregson

1992

Brit J Math & Statis

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Britirh Jaurnul tit Murhcmuric.ol und SruIi\liiid P\whnioy). 11992) 4). 1Jl-195 i Pri~ircd in t i r d Briruin 19Y2 The British P\ycholcr~ical SwielyThe details of response distributions in the method of limits are known to violate both simple assumptions of monotone increasing stimulussresponse functions, and the derived idea that change points on such functions, those points where sensory phenomenology changes abruptly, are strictly ordered correspondingly. The predictions of nonlinear psychophysics include mappings from responses back onto stimuli which are not single-valued. Such predictions imply that outliers to change point distributions expressed in physical units will exist under some conditions. The detection o f outliers is tractable from a Bayesian approach. An experiment using fractionation of perceived brightness by continuous response adjustment is used as an illustrative example both for theory and for statistical analyses.

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“…Its properties within a stable region have been shown to reproduce a diversity of empirical phenomena which are of interest to experimental psychologists. Its behaviour in the neighbourhoods of its higher attractors has been explored more recently (Campbell & Gregson 1990). These regions are of potential interest because they may be interpreted as evidence of system dynamics where there is sensory overload and a predictable breakdown of normal functioning.…”

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Dead men tell odd simple tales!

Gregson¹

1991

Behav Brain Sci

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Julia sets for the gamma recursion in nonlinear psychophysics

Abstract: Julia sets for the map z ~ alz -ie)(1 -zXz + ie) are illustrated for some attractors of interest. This work extends previous analyses of the cubic complex polynomial and considers dynamics in regions which may be associated with the modelling of the results of overload in sengory inputs.

Cited by 8 publications

References 10 publications

Similarities of low-dimensional chaotic auditory attractor sequences to quasirandom noise

Similarities of low-dimensional chaotic auditory attractor sequences to quasirandom noise

The psychophysical method of limits: What actually happens in a nonlinear context?

Dead men tell odd simple tales!

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