Some elements for a history of the dynamical systems theory

Letellier, Christophe; Abraham, Ralph; Shepelyansky, Dima L.; Rössler, Otto E.; Holmes, Philip; Lozi, René; Glass, Leon; Pikovsky, Arkady; Olsen, Lars Folke; Tsuda, Ichiro; Grebogi, Celso; Parlitz, Ulrich; Gilmore, Robert; Pecora, Louis M.; Carroll, Thomas L.

doi:10.1063/5.0047851

Cited by 17 publications

(11 citation statements)

References 155 publications

(180 reference statements)

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“…Several other papers have already been published, usually supported by a huge number of relevant works cited therein. For example, some elements for a history of dynamical systems theory are provided in a paper having the same title [186]. Pioneering work [187] gives a nice explanation of nonlinear dynamics observed within higher-order oscillatory circuits, both autonomous and driven.…”

Section: Discussionmentioning

confidence: 99%

Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example

Petržela

2022

Mathematics

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This paper strives to achieve a comprehensive review of chaos in analog circuits and lumped electronic networks. Readers will be guided from the beginning of the investigations of simple electronic circuits to the current trends in the research into chaos. The author tries to provide the key references related to this issue, including papers describing modern numerical algorithms capable of localizing chaotic and hyperchaotic motion in complex mathematical models, interesting full on-chip implementations of chaotic systems, possible practical applications of entropic signals, fractional-order chaotic systems and chaotic oscillators with mem-elements.

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Section: Discussionmentioning

confidence: 99%

Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example

Petržela

2022

Mathematics

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“…However, while the laws of physics enable us to predict the interactions of dynamic systems comprising just two bodies, the introduction of just a third gives rise to complex interactions that render systems chaotic and unpredictable, in the long term at least (Gowers et al, 2008). Poincare's proof of this in 1890 led to the development of chaos theory (Letellier et al, 2021), of which the butterfly effect (Lorenz, 1963) is possibly the best-known concept. This describes how very small changes in one part of a complex system may have very large effects elsewhere.…”

Section: Complexity and Complex Adaptive Systemsmentioning

confidence: 99%

An Evolutionary Theory of Values

Griffiths¹,

Thomas²,

Dyer³

2023

Preprint

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Brains, individuals, humanity, and the universe in which they emerged are complex adaptive systems. Driven by common motivations related to the second law of thermodynamics, their dynamism is regulated by localised cooperation and competition between components. Cooperation is associated with energy conservation and stability, competition with energy loss and disorder. All systems can be evaluated in terms of their propensity to change and the interplay of cooperation and competition between components. The Schwartz system of values, which provides a parsimonious description of the human motivational system, has axes that describe just these characteristics. We argue that its structure evolved to represent a pre-existing universal equivalent. The motives of stable, energy-conserving pre-biotic systems can be attributed to the equivalent of its conservation/cooperation quadrant. With the evolution of organisms, subordinate competitive motives were integrated to facilitate dynamism. Pro-change motives subsequently emerged in intelligent organisms, to be supplemented by widely framed and consciously cooperative motives in humanity. We present experimental and other evidence showing that human decision-making exhibits linear, hierarchical patterns of values-based influence consistent with this evolutionary sequence. The implications of this on established psychological theory, and the potential insights it offers in relation to personal, organizational, and societal development are discussed.

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“…Conducted in the spirit of recent works by Banaszak et al, (2015Banaszak et al, ( , 2019, this study is devoted to the dynamics of spatial interactions generated by urban units included in a regular hexagonal CPT pattern (see also : Allen, 1997;Allen and Sanglier, 1979;Allen, Sanglier, 1981;Alligood et al, 1996;Bandt and Mekhontsev, 2018;Chen, 2011;Dzięcielski et al, 2021;Fujita et al 2001;Ikeda et al, 2014Ikeda et al, , 2017Letellier et al, 2021;White, 1977). It seeks to develop a comprehensive dynamic 'umbrella' methodology in which prevailing static CPT approaches can be positioned and combined from a unitary Newtonian motion perspective on hierarchical systems.…”

Section: Aims and Scopementioning

confidence: 99%

Fractal Dimension Complexity of Gravitation Fractals in Central Place Theory

Banaszak

Górnisiewicz

Nijkamp³

et al. 2022

Preprint

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Settlement centers of various types, including cities, produce basins of attraction whose shape can be regular or complexly irregular (from the point of view of geometry). It depends, among others, on the properties of the space surrounding a city. Central Place Theory (CPT), a well-known concept in economic geography and spatial economics, explains the spatial structure inherent in distinct economic orders of urban centers in geographical space. In Christaller’s original formulation the fundamental architecture of CPT is based on a static approach which focuses on the optimal division of space around urban centers, while following pre-specified plausible rules of supplying them with goods from other places. Conducted in the spirit of recent works, this study demonstrates that by introducing a dynamic approach to space and including an equation of motion and space resistance, a dramatic change of the idealized static CPT image occurs. As a result of the interplay of gravitational forces, basins of attraction arise around cities, whose boundaries are fractals. This paper provides a wealth of spatial fractal complex images that question the relevance and validity of traditional CPT.

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Some elements for a history of the dynamical systems theory

Cited by 17 publications

References 155 publications

Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example

Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example

An Evolutionary Theory of Values

Fractal Dimension Complexity of Gravitation Fractals in Central Place Theory

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