Simple mathematical models with very complicated dynamics
Abstract
First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hiearchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.
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The Lorenz equation is a nonlinear chaotic system that exhibits complex dynamic behavior.
“The Lorenz equation is a nonlinear chaotic system that exhibits complex dynamic behavior.”
From Chaos-Based Image Encryption: Review, Application, and Challenges (2023) · cites this paper as [3]
The Lorenz equation is characterized by having a chaotic attractor with a shape that resembles a butterfly, and it is particularly known for its sensitivity to initial conditions.
“The Lorenz equation is characterized by having a chaotic attractor with a shape that resembles a butterfly, and it is particularly known for its sensitivity to initial conditions.”
From Chaos-Based Image Encryption: Review, Application, and Challenges (2023) · cites this paper as [3]
In 1976, Robert M. May proposed the Logistic map in an article, which was studied in depth by M.J. Feigenbaum, who proposed the universality of this map in 1978.
“In 1976, Robert M. May [3] proposed the Logistic map in an article, which was studied in depth by M.J. Feigenbaum, who proposed the universality of this map in 1978.”
From Chaos-Based Image Encryption: Review, Application, and Challenges (2023) · cites this paper as [3]
Despite being unpredictable, chaotic systems have recognizable features.
“Despite being unpredictable, chaotic systems have recognizable features (May 1976; Werndl 2009).”
From Religion as a Complex and Dynamic System (2013) · cites this paper as May 1976
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- Authors
- May, R. M.
- Affiliations
- Princeton University (US)
- Corresponding author
- May, Robert M.
- Venue
- Nature
- Published
- 1976-06-10
- Volume
- 261
- Issue
- 5560
- Pages
- 459-467
- ISSN
- 0028-0836 (ISSN-L)
- 1476-4687
- OpenAlex
- W2102892532
- Cited by
- 7,001
- References
- 0
- Citations by year
- 2026: 224 · 2025: 315 · 2024: 277 · 2023: 253 · 2022: 237 · 2021: 306 · 2020: 268 · 2019: 223 · 2018: 226 · 2017: 203 · 2016: 236 · 2015: 209 · 2014: 218 · 2013: 222 · 2012: 186
- FWCI
- 29.83
- Type
- journal article
- Language
- en
- Update notices
- correction (1976-07-15)
- Open access
- closed
- Locations
- Nature (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
- Health Sciences › Medicine › Public Health, Environmental and Occupational Health › Mathematical and Theoretical Epidemiology and Ecology Models
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Equations and Dynamical Systems
- Keywords
- first-order difference equations, discrete dynamical systems, chaotic dynamics, bifurcation, stable cycles, deterministic chaos, trajectory structure, nonlinear dynamics
- MeSH terms
- Diffusion, Mathematics, Models, Theoretical, Population Dynamics
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