Simple mathematical models with very complicated dynamics

May, Robert M.

doi:10.1007/978-0-387-21830-4_7

Cited by 308 publications

(353 citation statements)

References 21 publications

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“…On the other hand, a complicated piece of medical technology, such as a positron emission tomography scanner, will obviously not survive removal of a major component. The behaviour of a chaotic system appears random, but is generated by simple, non-random, deterministic processes: the complexity is in the dynamical evolution (the way the system changes over time driven by numerous iterations of some very simple rule), rather than the system itself 59 – 60. …”

Section: Dynamical Systemsmentioning

confidence: 99%

“…Such properties are known as emergent properties . In this way it is possible to have an upward (or generative ) hierarchy of such levels, in which one level of organisation determines the level above it, and that level then determines the features of the level above it 59. Emergent properties may also be universal or multiply realisable in the sense that there are many diverse ways in which the same emergent property can be generated.…”

Section: Chaos Versus Complexitymentioning

confidence: 99%

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A simple guide to chaos and complexity

Rickles

Hawe²,

Shiell³

2007

Journal of Epidemiology & Community Health

335

293

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Section: Dynamical Systemsmentioning

confidence: 99%

Section: Chaos Versus Complexitymentioning

confidence: 99%

A simple guide to chaos and complexity

Rickles

Hawe²,

Shiell³

2007

Journal of Epidemiology & Community Health

335

293

View full text Add to dashboard Cite

“…noninvertible maps: (1) Logistic map [40]; (2) Sine map [41]; (3) Tent map [42]; (4) Linear congruential generator [43]; (5) Cubic map [44]; (6) Ricker’s population model [45]; (7) Gauss map [46]; (8) Cusp map [47]; (9) Pinchers map [48]; (10) Spence map [49]; (11) Sine-circle map [50].…”

Section: Methodsmentioning

confidence: 99%

Distinguishing Noise from Chaos: Objective versus Subjective Criteria Using Horizontal Visibility Graph

et al. 2014

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A recently proposed methodology called the Horizontal Visibility Graph (HVG) [Luque et al., Phys. Rev. E., 80, 046103 (2009)] that constitutes a geometrical simplification of the well known Visibility Graph algorithm [Lacasa et al., Proc. Natl. Sci. U.S.A. 105, 4972 (2008)], has been used to study the distinction between deterministic and stochastic components in time series [L. Lacasa and R. Toral, Phys. Rev. E., 82, 036120 (2010)]. Specifically, the authors propose that the node degree distribution of these processes follows an exponential functional of the form , in which is the node degree and is a positive parameter able to distinguish between deterministic (chaotic) and stochastic (uncorrelated and correlated) dynamics. In this work, we investigate the characteristics of the node degree distributions constructed by using HVG, for time series corresponding to chaotic maps, 2 chaotic flows and different stochastic processes. We thoroughly study the methodology proposed by Lacasa and Toral finding several cases for which their hypothesis is not valid. We propose a methodology that uses the HVG together with Information Theory quantifiers. An extensive and careful analysis of the node degree distributions obtained by applying HVG allow us to conclude that the Fisher-Shannon information plane is a remarkable tool able to graphically represent the different nature, deterministic or stochastic, of the systems under study.

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“…It is similar to the classical logistic oscillator (May 1976) but being cubic can take positive or negative forms governed by the sign of p, providing so called P and N-oscillators. These oscillators were inspired by the cubic voltage-current relationships that have been proposed to account for some properties of spiking neurons including bursting (Hindmarsh and Rose 1982;.…”

Section: Oscillatorsmentioning

confidence: 98%

Discrimination of complex form by simple oscillator networks

Nagai

Taylor

Loh

et al. 2009

Network: Computation in Neural Systems

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Natural images are rich in higher order spatial correlations. Brain scanning, psychophysics and electrophysiology indicate that humans are sensitive to these image properties. A useful tool for exploring this sense is the set of isotrigon textures. Like natural images these textures have low dimensionality relative to random images, but like random images contain no average structure in their first to third order correlation functions. Thus, the structured appearance of these textures results from higher order correlations. One way to generate the higher order products inherent in higher order correlations is recursive nonlinear processing. We therefore decided to examine if very small oscillator networks could produce a profile of activity that matches human isotrigon discrimination performance across 53 isotrigon texture types. Human performance was measured in 23 subjects. The two best network types found contained as few as 4 oscillators. The input oscillators are of a novel cubic form and the final readout oscillator was a logistic oscillator. Mean readout oscillator activity matched human performance reasonably well even though the network parameters were fixed for all 53 texture types. Overall it appears that relatively simple, short range, and biologically plausible, recursive processing could provide the basis for discrimination of complex form.

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Simple mathematical models with very complicated dynamics

Cited by 308 publications

References 21 publications

A simple guide to chaos and complexity

A simple guide to chaos and complexity

Distinguishing Noise from Chaos: Objective versus Subjective Criteria Using Horizontal Visibility Graph

Discrimination of complex form by simple oscillator networks

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